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Totally positive kernels, Pólya frequency functions, and their transforms

Lookup NU author(s): Professor 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, Polya frequency functions, and Polya 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.

Publication metadata

Author(s): Belton A, Guillot D, Khare A, Putinar M

Publication type: Article

Publication status: Published

Journal: Journal d'Analyse Mathematique

Year: 2023

Volume: 150

Pages: 83–158

Online publication date: 05/01/2023

Acceptance date: 24/11/2021

ISSN (print): 0021-7670

ISSN (electronic): 1565-8538

Publisher: Magnes Press

URL: https://doi.org/10.1007/s11854-022-0259-7

DOI: 10.1007/s11854-022-0259-7

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