The Lévy-Khinchine type representations and generalized functions on Bernstein spaces

We consider the class of all univariate probability laws whose characteristic functions do not have zeroes on the real line. This class contains the fundamental family of infinitely divisible distributions and its new extension, the set of rational-(quasi-)infinitely divisible laws. We prove that this class is dense in the family of all probability laws with respect to the total variation distance. We show that characteristic functions corresponding to this class admit Lévy-Khinchine type representations. In these formulas, the generalized spectral functions (the regularized Lévy measures) are special continuous linear functionals on the union of well known Bernstein spaces of entire functions of exponential type which are additionally bounded on the real line. We apply this spectral functions in criteria of weak convergence of distributions within the class under study. All necessary analytical constructions are proposed.