Totally positive kernels, Polya frequency functions, and their transforms

TL;DR

This paper classifies the functions that preserve total non-negativity and total positivity in kernels, extending classical theorems and analyzing structured kernels using harmonic analysis and matrix completion techniques.

Contribution

It provides a comprehensive classification of composition operators preserving total positivity and non-negativity for various kernel classes, extending Whitney's density theorem.

Findings

Composition operators preserve total non-negativity only if the function is constant or linear.

Symmetric kernels have similar preservation properties under composition.

The results extend classical theorems and include analysis of structured kernels like Hankel and Pólya frequency functions.

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, P\'olya frequency functions, and P\'olya 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.