Hirschman-Widder densities
arXiv:2101.02129 · doi:10.1016/j.acha.2022.04.002
Abstract
Hirschman and Widder introduced a class of Pólya 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 Pólya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a Pólya 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.
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