Let G be a separable locally
compact group; ρ, a positive definite function; M(G), the set of all finite Radon
measures; and
Let Hρ be the Hilbert space obtained by completing M(G)∕Nρ. Similarly define Hσ
as the Hilbert space corresponding to another positive definite function σ. ρ and σ
are said to be equivalent (symbolically ρ ∼ σ) if there is an equivalence operator T
from Hρ to Hσ which is induced by the identity operator on M(G); i.e. a linear
homeomorphism from Hρ onto Hσ such that 1 −T∗T is Hilbert-Schmidt. Theorem 1
and Theorem 2 give necessary and sufficient conditions for ρ ∼ σ in terms of the
unitary representations of G induced by ρ and σ. We discuss group invariant
positive definite functions on X × X where X is a homogeneous space, and
generalize Theorem 1 and 2 accordingly. The notion of equivalence operators
comes exactly from Gaussian stochastic processes (cf. J. Feldman [4]). Some
statistical applications will be discussed in a separate paper later in the
year.
