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

Preservers of totally positive kernels and Polya frequency functions