Preservers of totally positive kernels and Polya frequency functions

TL;DR

This paper unifies the understanding of fractional powers and polynomial maps that preserve totally positive structures, revealing a fundamental spectral separation and extending classical results with insights from probability and group theory.

Contribution

It provides a comprehensive classification of structure-preserving transforms for totally positive matrices and kernels, integrating classical and modern mathematical concepts.

Findings

Discovered a spectral separation between discrete and continuous spectra of fractional powers.

Classified polynomial and fractional transforms preserving total positivity.

Connected classical results with modern probability and group theory frameworks.

Abstract

Fractional powers and polynomial maps preserving structured totally positive matrices, one-sided Polya frequency functions, or totally positive kernels are treated from a unifying perspective. Besides the stark rigidity of the polynomial transforms, we unveil an ubiquitous separation between discrete and continuous spectra of such inner fractional powers. Classical works of Schoenberg, Karlin, Hirschman, and Widder are completed by our classification. Concepts of probability theory, multivariate statistics, and group representation theory naturally enter into the picture.