Totally positive kernels, Pólya frequency functions, and their transforms

Belton, Alexander and Guillot, Dominique and Khare, Apoorva and Putinar, Mihai (2023) Totally positive kernels, Pólya frequency functions, and their transforms. Journal d'Analyse Mathématique, 150 (1). pp. 83-158. ISSN 0021-7670

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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.