[Paper Review] Post-composition transforms of totally positive kernels

Alexander C. R. Belton, Dominique Guillot+2 more|arXiv (Cornell University)|Jun 29, 2020

Mathematical Analysis and Transform Methods37 references4 citations

TL;DR

This paper classifies post-composition functions that preserve total non-negativity and total positivity in kernels across various classes, showing such functions must be linear (or constant for total non-negativity). The results stem from matrix completion theorems, an extension of Whitney's density theorem via discrete convolution with modulated Gaussians, and harmonic analysis of structured kernels like continuous Hankel and Polya frequency families.

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.

Motivation & Objective

• To classify functions that preserve total non-negativity and total positivity when composed post-hoc with kernels on arbitrary domains.
• To extend A.M. Whitney's density theorem from finite domains to subsets of the real line using discrete convolution with modulated Gaussian kernels.
• To analyze preservers of structured totally non-negative and totally positive kernels, including continuous Hankel kernels, Polya frequency functions, and sequences.
• To establish the rigid structural constraints on post-composition transforms acting on infinite sets by unifying results from specialized kernel families.
• To provide a comprehensive classification of such transforms by combining results from matrix completion and harmonic analysis techniques.

Proposed method

• Utilizes matrix completion theorems to characterize the structure of kernels preserved under post-composition.
• Applies discrete convolution with modulated Gaussian kernels to extend Whitney's density theorem to subsets of the real line.
• Employs tools from harmonic analysis to study preservers of continuous Hankel kernels on intervals and Polya frequency functions.
• Analyzes Polya frequency sequences using generating function and moment-based techniques to identify structural constraints.
• Combines results from multiple specialized kernel families to derive a unified classification for infinite-domain kernels.
• Relies on functional analytic methods to characterize the necessary and sufficient conditions on post-composition functions.

Experimental results

Research questions

• RQ1 Which functions preserve total non-negativity when composed post-hoc with kernels defined on arbitrary domains?
• RQ2 Which functions preserve total positivity under the same post-composition operation?
• RQ3 How can Whitney's density theorem be extended from finite domains to subsets of the real line using convolution with modulated Gaussians?
• RQ4 What are the structural constraints on post-composition transforms of continuous Hankel kernels on intervals?
• RQ5 How do harmonic analysis tools characterize the preservers of Polya frequency functions and sequences?

Key findings

• A post-composition function preserves total non-negativity in kernels if and only if it is constant or linear.
• For total positivity, the only such functions are linear functions, excluding constants.
• The extension of Whitney's density theorem to subsets of the real line is achieved via discrete convolution with modulated Gaussian kernels.
• The structure of post-composition transforms on infinite sets is rigidly constrained by the combined analysis of Hankel, Polya frequency, and sequence families.
• Matrix completion results provide foundational constraints that underlie the classification of preserving functions.
• Harmonic analysis tools successfully identify and characterize the functional forms that preserve structured totally positive and non-negative kernels.