Hirschman-Widder densities
Abstract
Hirschman and Widder introduced a class of P\'olya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a P\'olya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a P\'olya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.
Cite
@article{arxiv.2101.02129,
title = {Hirschman-Widder densities},
author = {Alexander Belton and Dominique Guillot and Apoorva Khare and Mihai Putinar},
journal= {arXiv preprint arXiv:2101.02129},
year = {2022}
}
Comments
32 pages, no figures. Numerous small additions, including Proposition 2.9, as well as Section 3 and other remarks connecting Hirschman-Widder densities to orbital integrals. Final version, to appear in Applied and Computational Harmonic Analysis