Home > Research > Publications & Outputs > Totally positive kernels, Pólya frequency funct...

Research

Electronic data

  • TP-Polya_revised

    Rights statement: 12m

    Accepted author manuscript, 757 KB, PDF document

    Embargo ends: 1/01/50

    Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License

Links

Keywords

View graph of relations

Totally positive kernels, Pólya frequency functions, and their transforms

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Forthcoming

Standard

Totally positive kernels, Pólya frequency functions, and their transforms. / Belton, Alexander; Guillot, Dominique; Khare, Apoorva et al.
In: Journal d'Analyse Mathématique, 24.11.2021.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Belton, A, Guillot, D, Khare, A & Putinar, M 2021, 'Totally positive kernels, Pólya frequency functions, and their transforms', Journal d'Analyse Mathématique. <https://arxiv.org/abs/2006.16213>

APA

Belton, A., Guillot, D., Khare, A., & Putinar, M. (in press). Totally positive kernels, Pólya frequency functions, and their transforms. Journal d'Analyse Mathématique. https://arxiv.org/abs/2006.16213

Vancouver

Belton A, Guillot D, Khare A, Putinar M. Totally positive kernels, Pólya frequency functions, and their transforms. Journal d'Analyse Mathématique. 2021 Nov 24.

Author

Belton, Alexander ; Guillot, Dominique ; Khare, Apoorva et al. / Totally positive kernels, Pólya frequency functions, and their transforms. In: Journal d'Analyse Mathématique. 2021.

Bibtex

@article{03a6dd111a994244bf47f4a603f7cca4,
title = "Totally positive kernels, P{\'o}lya frequency functions, and their transforms",
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{\textquoteright}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, P{\'o}lya frequency functions, and P{\'o}lya 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.",
keywords = "totally non-negative kernel, totally positive kernel, totally non-negative matrix, totally positive matrix, entrywise transformation, completion problem, P´olya frequency function, P´olya frequency sequence",
author = "Alexander Belton and Dominique Guillot and Apoorva Khare and Mihai Putinar",
year = "2021",
month = nov,
day = "24",
language = "English",
journal = "Journal d'Analyse Math{\'e}matique",
issn = "0021-7670",
publisher = "Springer",

}

RIS

TY - JOUR

T1 - Totally positive kernels, Pólya frequency functions, and their transforms

AU - Belton, Alexander

AU - Guillot, Dominique

AU - Khare, Apoorva

AU - Putinar, Mihai

PY - 2021/11/24

Y1 - 2021/11/24

N2 - 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.

AB - 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.

KW - totally non-negative kernel

KW - totally positive kernel

KW - totally non-negative matrix

KW - totally positive matrix

KW - entrywise transformation

KW - completion problem

KW - P´olya frequency function

KW - P´olya frequency sequence

M3 - Journal article

JO - Journal d'Analyse Mathématique

JF - Journal d'Analyse Mathématique

SN - 0021-7670

ER -