Hirschman-Widder densities

Belton, Alexander and Guillot, Dominique and Khare, Apoorva and Putinar, Mihai (2022) Hirschman-Widder densities. Applied and Computational Harmonic Analysis, 60. pp. 396-425. ISSN 1063-5203

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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.