  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.
