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arXiv · 2006.16213

Totally positive kernels, Polya frequency functions, and their transforms

Alexander Belton·Dominique Guillot·Apoorva Khare·Mihai Putinar
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Abstract

The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A.M. Whitney's density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Pólya frequency functions, and Pólya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.

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Alexander Belton, Dominique Guillot, Apoorva Khare, Mihai Putinar. 2021-12-07. Totally positive kernels, Polya frequency functions, and their transforms. https://doi.org/10.1007/s11854-022-0259-7

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