Preservers of totally positive kernels and Polya frequency functions
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.
Cite
@article{arxiv.2110.08206,
title = {Preservers of totally positive kernels and Polya frequency functions},
author = {Alexander Belton and Dominique Guillot and Apoorva Khare and Mihai Putinar},
journal= {arXiv preprint arXiv:2110.08206},
year = {2022}
}
Comments
25 pages, no figures. This is an announcement of some of the results in a sequence of three closely related recent papers arXiv:2006.16213, arXiv:2008.05121, and arXiv:2101.02129. Final version, published in Mathematics Research Reports