An integral representation for a class of positive definite functions

Abstract

is satisfied for all k, k' e K and x e G then ~b is said to be bi-invariant with respect to K. Now let B be the set of all elementary normalized positive definite functions on G, bi-invariant with respect to K, endowed with the weak-* topology induced by LI(G). (See [2], {} 30, for definitions and elementary properties of positive definite functions.) It is known that if G is unimodular, and K is a compact subgroup and if a certain subalgebra of LI(G ) is commutative, then corresponding to each bi-invariant positive definite function ~b on G there is a Radon measure/~ on B such that