Preservers of totally positive kernels and Pólya frequency functions
Mathematics Research Reports, Volume 3 (2022), pp. 35-56.

Fractional powers and polynomial maps preserving structured totally positive matrices, one-sided Pólya 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.

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DOI: 10.5802/mrr.12
Classification: 15B48,  15A15,  39B62,  42A82,  44A10,  47B34
Keywords: Totally non-negative function, totally positive function, totally non-negative kernel, totally positive kernel, totally non-negative matrix, totally positive matrix, entrywise transformation, Pólya frequency function, Pólya frequency sequence, Hirschman–Widder density, exponential random variable, spherical function, orbital integral, multivariate statistics.
Alexander Belton 1; Dominique Guillot 2; Apoorva Khare 3; Mihai Putinar 4

1 Department of Mathematics and Statistics, Lancaster University, Lancaster, UK
2 University of Delaware, Newark, DE, USA
3 Department of Mathematics, Indian Institute of Science, Bangalore, India; and Analysis and Probability Research Group, Bangalore, India
4 University of California at Santa Barbara, CA, USA; and Newcastle University, Newcastle upon Tyne, UK
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Alexander Belton; Dominique Guillot; Apoorva Khare; Mihai Putinar. Preservers of totally positive kernels and Pólya frequency functions. Mathematics Research Reports, Volume 3 (2022), pp. 35-56. doi : 10.5802/mrr.12. https://mrr.centre-mersenne.org/articles/10.5802/mrr.12/

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