Functions operating on positive definite matrices and a theorem of Schoenberg

Abstract

We prove that the set of all functions f : [ − 1 , 1 ] → [ − 1 , 1 ] f:\,[ - 1,\,1] \to [ - 1,\,1] operating on real positive definite matrices and normalized such that f ( 1 ) = 1 f(1)\, = \,1 , is a Bauer simplex, and we identify its extreme points. As an application we obtain Schoenberg’s theorem characterising positive definite kernels on the infinite dimensional Hilbert sphere.