Belton, A and Guillot, D and Khare, A and Putinar, M (2023) Totally positive kernels, Pólya frequency functions, and their transforms. In: Journal d'Analyse Mathematique .
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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. © 2022, The Hebrew University of Jerusalem.
| Item Type: | Journal Article |
|---|---|
| Publication: | Journal d'Analyse Mathematique |
| Publisher: | Hebrew University Magnes Press |
| Additional Information: | The copyright for this article belongs to the Authors. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 02 Feb 2023 10:21 |
| Last Modified: | 02 Feb 2023 10:21 |
| URI: | https://eprints.iisc.ac.in/id/eprint/79784 |
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