Bounds for Orthogonal Expansion Coefficients via Weighted Primitives

Introduction

An exact formula for a Fourier coefficient requires information about the whole function. In applications, such information may be unavailable, and the full curve may be impossible to prescribe. It is then useful to replace exact coefficient calculation by inequalities involving quantities that can be estimated or controlled, such as total variation. In this sense, the problem is one of Fourier analysis in inequalities.

The motivation comes from the control of selected harmonics of the circular distribution of velocity upstream of a propeller. In the formulation of [1], ideal curves have selected harmonics absent, and admissible departures from those curves are constrained so that the corresponding harmonics remain small. The mathematical requirement is an estimate connecting the variation of a curve, or of its departure from an ideal curve, with its expansion coefficients. Section 10 develops this use of the inequalities through explicit admissible corridors. The construction of ideal curves and their application to propeller inflow were introduced in [15] and subsequently discussed and illustrated in [16, 17]. A separate paper will present a fuller account and develop a broader formulation of the controlled-distortion method. We use the term harmonics for the trigonometric components of the periodic inflow; for a general orthogonal system we refer to expansion coefficients or components.

The Riemann–Lebesgue theorem gives convergence of Fourier coefficients to zero. Additional assumptions on variation or smoothness yield quantitative decay estimates, including the classical $1/n$ bound for periodic functions of bounded variation; see Taibleson [4] and Edwards [6]. Here bounded variation is the regularity assumption, and the aim is to identify what features of a specified orthogonal system permit analogous estimates. The explicit numerator is essential for this purpose: an asymptotic order alone does not specify a design tolerance at a selected harmonic.

Integration by parts gives the estimate $|I_n(f)| \le V(f)M_n$. The factor $M_n$ is the smallest constant for which this inequality holds for every function of bounded variation. It depends only on the chosen basis function, its normalization, the weight, and the interval, and can therefore be calculated independently of the function being expanded. The zeros describe the oscillatory structure of the basis element, while the weighted primitive measures the accumulated balance of its positive and negative contributions.

The connection between bounded variation and primitives of general orthonormal systems predates the present investigation. Tsagareishvili [7] announced a necessary and sufficient condition for $O(1/n)$ coefficient decay for every BV function, under uniform boundedness and zero-mean assumptions on the orthonormal system. His later paper [8 Theorems 5 and 5’] gives grid and continuous primitive formulations. The 1981 note states the results without proofs. On $[0,1]$, if $\|\phi_n\|_\infty\le B$, the grid quantity $K_n=\max_{0\le i\le n}\left|\int_0^{i/n}\phi_n(t)\,dt\right|$ satisfies

Thus the two primitive conditions have the same $O(1/n)$ content under those assumptions. These results concern bounds for each fixed BV function, with the implied constant allowed to depend on that function. Here the exact constant for a prescribed variation bound is kept explicit. The zero-mean kernel argument also applies when the normalized basis functions are not uniformly bounded. Gogoladze and Tsagareishvili [5, 9] provide further relevant results for general orthonormal systems.

The integration by parts estimate is a classical consequence of the BV bound for Stieltjes integrals; see, for example, Dragomir [10]. Closely related Legendre coefficient and Gegenbauer primitive estimates appear in Wang [11], and weighted Gegenbauer estimates go back to Durand [12]. Xiang and Liu [13 Theorem 1 with $k=0] give precisely the Jacobi coefficient bound $V(f)M_n/h_n$, with an explicit weighted polynomial expression for $M_n$. Their Remark 1 also locates the primitive maxima at zeros of the original polynomial. Their uniform weighted estimates yield the Jacobi orders compared here. Zhang, Xiang, and Kong [14 Theorem 2.4] give coefficient decay for Laguerre expansions of piecewise functions, including the $O(n^{-3/4})$ estimate for the ordinary Laguerre step-function example below. The present treatment organizes these classical bounds and their consequences into a unified comparison through $M_n$, explains the geometry by integration between zeros, and converts shape restrictions into spectral guarantees in controlled distortion.

Notation and normalization

Let $I=[a,b]$ and let $w$ be a nonnegative integrable weight. Consider real orthogonal functions $p_0,p_1,\ldots$ with $p_0$ constant, and write

The coordinate coefficient and the orthonormal coefficient are, respectively,

The bounds for individual coefficients do not require the system to represent every function. Completeness is assumed only when we discuss approximation by the full expansion; it means that finite linear combinations of the basis functions are dense in $L^2(w)$.

The total variation is

We abbreviate this to $V(f)$ when the interval is understood. If $f$ is absolutely continuous, then $V(f)=\int_a^b\lvert f'(x)\rvert\,dx$. The partition definition also covers jumps and singular continuous variation. For a continuous curve with a finite partition into monotone arcs, variation is the sum of the absolute changes along those arcs, including the end arcs.

Throughout, $n\ge1$. For orthogonal polynomials, $p_n$ has degree $n$. Conventional normalizations are used unless stated otherwise. Constants denoted by $K$ may depend on the fixed parameters defining the polynomial family, but are independent of the degree $n$ and the function $f$. Their values may differ between estimates. For positive sequences, $u_n\asymp v_n$ means that there exist constants $c,C>0$ such that

for all sufficiently large $n$. Thus the sequences have the same asymptotic order.

The sharp primitive bound

The maximum is attained in the BV class. The same value is the supremum if $f$ is required to be continuous and to have variation at most one; in that smaller class it need not be attained. The threshold one is a normalization: for any $B > 0$, the corresponding maximum over $V(f) \le B$ is $B\|G\|_\infty$.

Proof. The primitive $G$ is absolutely continuous and satisfies $G(a) = G(b) = 0$. Stieltjes integration by parts, valid here even when $f$ has jumps, gives

The absolute value of the last integral is at most $\|G\|_\infty V(f)$. To prove sharpness, suppose $G$ is not identically zero and choose $c \in(a,b)$ with $|G(c)| = \|G\|_\infty$. The indicator function $f = \mathbf{1}_{[c,b]}$, equal to zero before $c$ and one after $c$, has variation one and satisfies

Thus equality is attained in the BV class. Replacing the jump by a monotone continuous ramp of width tending to zero preserves variation one and makes the integrals converge to $-G(c)$. The zero kernel case is immediate. This proves the theorem.

Apply Theorem 1 to $k = p_n w$. Orthogonality to constants gives zero total integral. With

we obtain the sharp bounds

In particular, a constant $C$ satisfying

for every $n \ge1$ and every BV function exists if and only if $\sup_{n\ge1} nM_n < \infty$. For a fixed normalization, the best such constant is $\sup_n nM_n$. This statement concerns a bound uniform in $f$. It does not claim that every fixed function has the worst possible coefficient behavior.

The same proof applies on $[0,\infty)$ when $k$ is integrable with zero integral, $f$ has finite total variation there, and the primitive satisfies $G(0) = 0$ and tends to zero at infinity. Such an $f$ is bounded. Integration by parts on finite intervals and passage to the limit justify the formula.

Integration between zeros and cumulative signed areas

For polynomials orthogonal with a positive weight on an interval, all $n$ zeros of the degree $n$ polynomial are simple and lie inside the interval [2 Section 18.2]. Thus the coefficient index is also the number of sign changes. If

the zeros divide the interval into $n+1$ sign intervals. The points $z_0$ and $z_{n+1}$ are the endpoints, not additional zeros. Define, for $0\leq k\leq n$,

On each interval $[z_k,z_{k+1}]$, the weighted polynomial $p_nw$ has a constant sign. For continuous $f$, the weighted mean value theorem therefore gives a point $\xi_k\in[z_k,z_{k+1}]$ such that

These points are ordered, $\xi_0\leq\cdots\leq\xi_n$. Summing the interval integrals gives

Since $\sum A_k=0$, discrete summation by parts gives

The variation bound enters through the ordered evaluation points:

Consequently,

Between successive zeros, $G_n$ is monotone. Since $G_n(a)=G_n(b)=0$, its largest absolute value is attained at an interior zero of $p_n$. Thus $\max_{0\leq k\leq n-1}|S_k|=M_n$. Division by $h_n$ recovers the coefficient estimate of Theorem 1 for continuous $f$. The Stieltjes proof covers the whole BV class, including discontinuous functions.

Equal absolute areas between zeros are unnecessary. The relevant quantity is the largest cumulative signed area. Counting zeros describes the oscillations, but does not alone bound this cumulative area.

Elementary family estimates

The cosine system and periodic trigonometric functions

For $p_n(x)=\cos(nx)$ on $[0,\pi]$ with $w=1$,

Consequently $|a_n(f)|\leq2V(f)/(\pi n)$, with the exact integral constant $1/n$.

On a full period $[0,2\pi]$, assume $f(0)=f(2\pi)$. The primitives $\sin(nx)/n$ and $-\cos(nx)/n$ give, by integration by parts,

The sine boundary term vanishes because the endpoint values of $f$ agree. Both coordinate coefficients, whose squared basis norms are $\pi$, are bounded by $V(f)/(\pi n)$. This is the periodic convention used in the motivating estimate [1]. Without periodic endpoint matching, the anchored sine primitive $(1-\cos(nx))/n$ instead gives a bound $2V(f)/n$ for the unnormalized integral. The boundary convention therefore belongs in the statement.

Chebyshev polynomials of the first kind

Let $p_n=T_n$ and $w(x)=(1-x^2)^{-1/2}$ on $[-1,1]$. With $x=\cos\theta$,

Hence $M_n=1/n$. Since $h_n=\pi/2$,

The substitution is monotone, so $V_{[0,\pi]}(f(\cos\theta))=V_{[-1,1]}(f)$. The result is the cosine estimate in a different variable.

Legendre polynomials

For $p_n=P_n$ and $w=1$, the standard identities in [2 Section 18.9] give

At any zero $z$ of $P_n$, the recurrence relation implies $(n+1)P_{n+1}(z)=-nP_{n-1}(z)$. Therefore

Using $|P_{n-1}(x)|\le1$ on $[-1,1]$, and the location of the extrema of $G_n$, we obtain

This convenient explicit bound is stronger than $V(f)/(nh_n)$, but is not asymptotically optimal. § 6 gives $M_n=O(n^{-3/2})$. Wang [11 Theorem 3.1, case $m=0$] uses the same Stieltjes argument and a sharper bound for $P_{n+1}-P_{n-1}$ to obtain an explicit $O(V(f)n^{-1/2})$ coordinate coefficient estimate. Our elementary estimate is included for its short zero-based proof, rather than as an improvement on that result.

Chebyshev polynomials of the second kind

For $p_n=U_n$ and $w(x)=\sqrt{1-x^2}$, use

Product to sum gives

Consequently

With $h_n=\pi/2$, this gives $|a_n(f)|\leq V(f)(n+1)/(n(n+2)h_n)$.

Chebyshev polynomials of the third kind

Use the convention

Here $h_n=\pi$. The transformed integrand is $2\cos(u/2)\cos((n+\frac{1}{2})u)=\cos(nu)+\cos((n+1)u)$, and hence

Thus $M_n\leq1/n+1/(n+1)<2/n$, giving $|a_n(f)|\leq2V(f)/(nh_n)$. The best constant uniform in $n$ is exactly 2. Indeed, at $\theta_n=\pi/(2n+1)$,

Together with the upper bound, this proves $nM_n\to2$. The fourth kind, with the reflected convention and reflected weight, has the same primitive norms by the change $x\mapsto-x$.

These bounds can also be recovered directly from classical cosine coefficient estimates. Set $F(\theta)=f(\cos\theta)$ and $c_j=(2/\pi)\int_0^\pi F(\theta)\cos(j\theta)\,\mathrm{d}\theta$. Product identities give, for $n\geq1$,

Applying $|c_j|\leq2V(f)/(\pi j)$ gives the same upper bounds. The weighted primitives make the associated sharp integral constant $M_n$ visible; the displayed upper inequalities themselves are elementary consequences of the classical cosine estimate.

Jacobi polynomials

Let $p_n = P_n^{(\alpha,\beta)}$ have the conventional Jacobi normalization and weight

The weighted derivative identity [2 formula 18.9.16], with its parameters shifted by one, gives

The bracketed function vanishes at both endpoints, so

Proof. We use the standard uniform Jacobi envelope, for fixed $A,B>0$,

This envelope follows from the uniform endpoint Bessel expansions in [2 formulas 18.15.6–18.15.8], together with their reflected versions and the interior Darboux expansion. It is a classical input to the proof, rather than an estimate derived here from orthogonality alone. To see the endpoint scaling, the small argument bound $J_A(u)=O(u^A)$ gives $O(n^A)$ for the polynomial when $\theta\le1/n$; for $\theta\ge1/n$, the bound $J_A(u)=O(u^{-1/2})$ gives the factor $n^{-1/2}\theta^{-A-1/2}$ away from the opposite endpoint. Fixed small degrees are absorbed into $K$.

Set $A=\alpha+1$, $B=\beta+1$. On $0\le\theta\le\pi/2$, the opposite endpoint factors are bounded. Since $1-\cos\theta$ is comparable to $\theta^2$,

For $0\le\theta\le1/n$, the right hand side is at most $Kn^{-\alpha-2}$. For $1/n\le\theta\le\pi/2$, it is bounded by $Kn^{-3/2}\theta^{\alpha+1/2}$. Its maximum there is $O(n^{-3/2})$ when $\alpha\ge-1/2$, and $O(n^{-\alpha-2})$ otherwise. Reflection gives the corresponding bounds involving $\beta$ on the other half of the interval. Combining the bounds proves the proposition.

The coefficient estimate is therefore

The conventional Jacobi squared norm satisfies $h_n \asymp n^{-1}$ for fixed parameters. Accordingly, this is a bound $O(V(f)n^{1-q})$ for coordinate coefficients and $O(V(f)n^{1/2-q})$ for orthonormal coefficients. For $\alpha,\beta\ge-1/2$, these become $O(V(f)n^{-1/2})$ and $O(V(f)n^{-1})$, respectively. The primitive rate must not be confused with the coordinate coefficient rate.

These estimates are consistent with the explicit BV bounds of Xiang and Liu [13]. Their Theorem 1, Eq. (14), at $k=0$ is exactly $\lvert a_n(f)\rvert\le V(f)M_n/h_n$ with the primitive displayed above. Their Theorem 4, Eq. (34), applied with parameters $\alpha+1,\beta+1>0$, bounds the weighted polynomial by $O(n^{-\min\{1/2,\alpha+1,\beta+1\}})$ and hence gives the same exponent $q$ for $M_n$. Their Remark 1, Eq. (17), identifies the zeros at which the maximum can be computed. The proof here spells out the endpoint and interior contributions in the notation used throughout this comparison.

Gegenbauer normalization

Restrict here to $\lambda>0$. Conventional Gegenbauer polynomials satisfy [2 formula 18.7.1]

where $(c)_n$ denotes the rising factorial. Both systems use the weight $(1-x^2)^{\lambda-1/2}$, so their primitives and their uniform norms satisfy

Since the corresponding Jacobi parameters exceed $-1/2$, Proposition 2 gives $M_n^{\mathrm{Jac}}=O(n^{-3/2})$. Multiplication by $d_n\asymp n^{\lambda-1/2}$ therefore gives

This order also has a matching lower bound. On a closed interior interval, the Darboux formula for the shifted Jacobi polynomial in the primitive has the form $n^{-1/2}A(\theta)\cos(n\theta+\gamma(\theta))+O(n^{-3/2})$, where $A$ is positive and bounded away from zero. The phase is affine in $\theta$ with slope $n+O(1)$. For every sufficiently large $n$, a point in that interval can be chosen where the cosine has magnitude one. The weight factors remain bounded away from zero. Multiplication by $d_n/n$ therefore gives a lower bound of order $n^{\lambda-2}$. Thus

It follows that the conventional primitive condition $M_n=O(1/n)$ holds exactly for $0<\lambda\le1$. Legendre and Chebyshev II occur at $\lambda=1/2$ and $\lambda=1$.

The same primitive can be written directly as

Durand’s weighted Gegenbauer bound [12 Eq. (19)] gives the explicit consequence

The ratio of gamma functions is of order $n^\lambda$, which again gives $O(n^{\lambda-2})$. Wang [11 Section 4.1] studies the corresponding generalized Gegenbauer–Gauss–Lobatto function, equal to $2(n+\lambda)G_n^{\mathrm{Geg}}$, and gives detailed maximum estimates. These results are direct predecessors of the primitive comparison used here.

The same Durand estimate is Lemma 2, Eq. (21), of Xiang and Liu [13]. Their Section 2.1 uses it to give explicit Gegenbauer coefficient bounds and extensions to derivatives of bounded variation. For the BV case considered here, their Jacobi Theorem 1 together with this lemma gives the displayed primitive bound after changing normalization. Their Theorem 3 is stated for derivative orders $k\ge1$; no extension of that theorem’s stated assumptions is needed for our $k=0$ comparison.

An explicit example makes the failure for larger parameters visible. At $\lambda=2$,

For $n=2m+1$, using $C_{2m}^{(3)}(0)=(-1)^m(m+1)(m+2)/2$,

These conventional primitives do not tend uniformly to zero. Compactness of the orthogonality interval alone is therefore insufficient for the primitive condition.

For orthonormal Gegenbauer polynomials the result is different. Since $h_n\asymp n^{2\lambda-2}$, their primitives have norm

This agrees with the corresponding orthonormal Jacobi system. Rescaling changes $M_n$ and the coordinate coefficients, while the ratio $M_n/\sqrt{h_n}$ is invariant under a nonzero rescaling of each basis member. Ordering still matters and is fixed here by degree. Parameters $-1/2<\lambda<0$ are outside the scope of this section.

Ordinary Laguerre polynomials

Let $p_n=L_n$ on $[0,\infty)$ with weight $e^{-x}$. Ordinary Laguerre polynomials are already orthonormal for this weight: $h_n=1$. Formula 18.9.24 in [2] yields

For each degree, this vanishes at zero and tends to zero at infinity. We show that its uniform norm is not $O(1/n)$.

For fixed $x>0$, the classical Laguerre expansion [2 formulas 18.15.14–18.15.16] gives

The phase increases without bound and its increments tend to zero. For each sufficiently large multiple of $2\pi$, select the first index for which the phase exceeds it. The overshoot tends to zero. Along the resulting subsequence the cosine tends to one, so $|G_n(x)|\ge c_xn^{-3/4}$. Consequently $nM_n$ is unbounded.

This gives a fixed BV counterexample to $1/n$ coefficient decay. Take $f(t)=\mathbf{1}_{[x,\infty)}(t)$, with $V(f)=1$. Orthogonality to constants and $h_n=1$ imply $a_n(f)=-G_n(x)$, which is not $O(1/n)$. Continuous ramps also rule out a $1/n$ inequality uniform over continuous functions of variation at most one, by Theorem 1. The fixed step example alone does not specify the optimal rate for each individual continuous function.

This step belongs to the class $z(t)(t-a)_{+}^{\mu}$ studied by Zhang, Xiang, and Kong [14]: take $z\equiv1$, $\mu=0$, and $a=x>0$. Their convention assigns the midpoint value $1/2$ at the jump; changing a value at one point leaves the coefficients unchanged. Their norm factor $\Gamma(n+\alpha+1)/n!$ also reduces to one when $\alpha=0$. Their Theorem 2.4, p. 6, gives, for the class defined in their Eq. (1),

In our step case this is $O(n^{-3/4})$, in agreement with the direct primitive calculation. The oscillatory formula above supplies the subsequence on which the $1/n$ bound fails. Thus this example illustrates an established decay phenomenon through a short exact primitive identity. The rate at a fixed jump location does not by itself determine $M_n$, which takes the supremum over all locations for each degree.

This argument establishes failure for ordinary Laguerre polynomials. It neither gives a global upper estimate $M_n=O(n^{-3/4})$ nor classifies all generalized Laguerre parameters; both would require additional analysis.

Comparison of the primitive estimates

The bounds below concern $M_n$, the sharp constant for the unnormalized weighted integral. To obtain a coordinate coefficient bound, multiply by $V(f)/h_n$.

Conventional system	Estimate for $M_n$
Cosine on $[0,\pi]$	$M_n = 1/n$
Chebyshev I	$M_n = 1/n$
Legendre	$M_n \le1/(n+1)$ and $M_n = O(n^{-3/2})$
Chebyshev II	$M_n \le(n+1)/(n(n+2)) < 1/n$
Chebyshev III	$M_n < 2/n$ and $nM_n \to2$
Jacobi with $\alpha,\beta> -1$	$M_n = O(n^{-q})$, $q = \min\{3/2,\alpha+2,\beta+2\} > 1$
Gegenbauer with $\lambda> 0$	$M_n \asymp n^{\lambda-2}$
Ordinary Laguerre	$M_n$ is not $O(1/n)$

The general coefficient inequality is sharp with $M_n$ itself. A convenient upper estimate for $M_n$, such as $1/(n+1)$ in the Legendre case, need not be the best numerical constant for each degree.

The numerical illustration below distinguishes a primitive norm from a fixed-point witness. In its upper panel the extrema of each primitive are evaluated at the polynomial zeros. In its lower panel only $|G_n(1)|$ is plotted for Laguerre, not the global maximum $M_n$. Since $|G_n(1)| \le M_n$, the growing subsequence of $n|G_n(1)|$ already rules out boundedness of $nM_n$. The calculations illustrate the analytic arguments; they are not used as proofs.

Controlled distortion and other consequences

Bounds from incomplete information

Suppose only $V(f) \le B$ is known. Theorem 1 gives $|a_n(f)| \le BM_n/h_n$ for every admissible function, without calculating its coefficients. Sharpness means that no smaller bound can be guaranteed from this variation information alone. A smaller bound may become possible if additional restrictions, such as monotonicity on prescribed subintervals or boundary conditions, are imposed.

Ideal curves and variation corridors

Let $f_\ast$ be an ideal curve and set $u=f-f_\ast$. Linearity gives

Fix a finite nonempty target set $\mathcal{T}$ and suppose $a_n(f_\ast)=0$ for every $n\in\mathcal{T}$. Given positive tolerances $\varepsilon_n$, put

Every curve in the admissible class

satisfies $\lvert a_n(f)\rvert\le\varepsilon_n$ simultaneously for all $n\in\mathcal{T}$. The ideal itself need not be physically attainable: the purpose is to identify attainable members of a class with guaranteed spectral behavior. Since constants do not affect the target coefficients or total variation, a prescribed mean may be imposed separately. If the target coefficients of $f_\ast$ are small rather than zero, the corresponding guarantee is $\lvert a_n(f)\rvert\le\lvert a_n(f_\ast)\rvert+\eta M_n/h_n$.

This is the mathematical use of the inequalities in controlled distortion [1]. Total variation is applied to the departure from the ideal, rather than only to the original flow curve. The result supplies sufficient spectral guarantees; whether a curve in this class can be produced by a realizable flow modification is a separate design question.

Geometric corridors and restrictions on extrema

A bound on maximum pointwise departure alone does not bound total variation. One useful additional restriction, employed in the controlled-distortion construction of [1], is a bound on the number of extrema of the departure $u=f-f_\ast$. Suppose $u$ is continuous and piecewise monotone, $\lvert u(x)\rvert\le\delta$, and there are at most $r$ interior turning points, with monotonicity between successive turning points and the endpoints. There are then at most $r+1$ monotone arcs. Each contributes at most $2\delta$ to the variation, so

For a nonconstant periodic departure with $r$ alternating extrema counted around the period and monotonicity between successive extrema, there are $r$ cyclic monotone arcs. In that case,

where $d$ is the full width of the corridor around the ideal curve. A constant departure has variation zero. These relations connect corridor width and the number of extrema with a guaranteed bound on selected coefficients. The extrema to be counted are those of $f-f_\ast$, not necessarily those of $f$.

For periodic inflow velocity $V(\varphi)$, let $V_*(\varphi)$ have zero sine and cosine coefficients at the target indices. If the periodic departure $u=V-V_*$ lies in a corridor of full width $d$ and has at most $r$ cyclic extrema as above, Section 5.1 gives, for the real Fourier coefficients,

Here $A_n$ and $B_n$ denote the cosine and sine coordinate coefficients in the usual real Fourier convention, with factor $1/\pi$. Write the amplitude of the $n$th real harmonic as $H_n=(A_n^2+B_n^2)^{1/2}$. In fact the same bound holds for this amplitude:

Indeed, for every phase $\psi$, the coefficient $A_n\cos\psi+B_n\sin\psi$ is the integral against $\cos(n\varphi-\psi)$ divided by $\pi$. Its periodic primitive has absolute value at most $1/n$, and maximizing over $\psi$ gives $H_n$. Thus $d\le\min_{n\in\mathcal{T}}(\pi n\varepsilon_n/r)$ is a sufficient corridor-width condition for $H_n(V)\le\varepsilon_n$ when $r>0$.

Choices of an ideal curve

The reference curve can be chosen to suit the design problem. A constant ideal removes every nonzero harmonic. A dynamic ideal retains a selected small set of major harmonics of the original curve, provided the target harmonics are excluded. An optimal ideal removes only the target sine and cosine components; it is the orthogonal projection onto the subspace where those components vanish and minimizes the unweighted mean-square departure on the period. A partial ideal coincides with the original outside prescribed sectors and satisfies the target harmonic conditions through modifications within those sectors. Modifications of peaks form a subclass of the partial construction. Existence under additional amplitude, sign, or physical restrictions must be examined in each application.

For all these choices, the same coefficient inequality turns restrictions on the departure into sufficient spectral guarantees. The construction of these ideal curves and their application to the propeller-flow problem were introduced in [15] and subsequently discussed and illustrated in [16, 17]. A separate paper will present a fuller account and develop a broader formulation of the controlled-distortion method. The present application isolates the mathematical step shared by them.

Truncated expansions

Assume the orthogonal system is complete in $L^2(w)$ and $f\in L^2(w)$. For $S_Nf=\sum_{n=0}^{N}a_n(f)p_n$, Parseval’s identity gives

The bound is useful when the right hand series converges. For conventional Jacobi polynomials, Proposition 2 and $h_n\asymp n^{-1}$ give

Indeed, $\sum_{n>N} n^{1-2q}=O(N^{2-2q})$ because $q>1$. In particular, the bound is $O(V(f)N^{-1/2})$ when $\alpha,\beta\ge-1/2$. These estimates provide a quantitative approximation guarantee, rather than a claim of an optimal error rate for every fixed function.

Scope and further questions

The general principle applies to any real integrable zero mean kernel, so orthogonality is sufficient for producing the kernels but is not required by Theorem 1. Orthogonality becomes important in identifying coefficients, interpreting their indices, and relating their squares to approximation error.

The remaining classification problem must specify the functions, their ordering, normalization, weight, and boundary conditions. One may study either $M_n$ in a prescribed conventional normalization or the normalization invariant quantity $M_n/\sqrt{h_n}$ for orthonormal coefficients. The latter permits direct comparisons between differently normalized descriptions of the same orthogonal system.

Further work may examine optimal constants for individual degrees, generalized Laguerre parameters, and systems of eigenfunctions arising from physical boundary value problems. For such systems, the natural index may be related to an eigenvalue or spatial scale rather than polynomial degree. The question remains concrete: how does the largest cumulative signed area depend on that natural index, and what coefficient guarantees follow for functions with bounded variation?

The family estimates presented here are established results or consequences of the classical identities and bounds cited above. In particular, [13] supplies the Jacobi BV coefficient inequality and the weighted estimates behind its decay orders, while [14] includes the Laguerre step-function class. The present organization makes the sharp numerator $V(f)M_n$, its geometric interpretation, and its use in admissible corridors explicit. This supplies a common mathematical description for coefficient estimates and the controlled-distortion application without asserting priority for the general inequalities.

Acknowledgments

This manuscript was developed in dialogue with ChatGPT (OpenAI), which provided substantial assistance in deriving and checking family estimates, analyzing normalization, searching for historical references, and drafting and organizing the text. This contribution extended beyond language editing. The investigation originated in the author’s earlier derivations of the trigonometric and Chebyshev I estimates and their use in controlled distortion [1]; this statement describes the development of the work, rather than a claim of priority over the classical coefficient theory. The author directed the investigation and examined the arguments during its development. Responsibility for verification of the final mathematical statements and references remains with the author.

References