Abstract:

I will discuss the theory of Polya frequency functions. These are totally nonnegative kernels which are at once connected to analysis, matrix theory, and probability. We will see examples and basic properties, and discuss Schoenberg’s classification of all such functions via the Laplace transform. I will also explain why the set of preservers of such functions is very rigid. If instead one studies powers preserving total positivity of finite order, this set is richer, and resembles sets of powers occurring in other areas (Berezin / Gindikin / Wallach sets).