Vol. 26, No. 1, 1968

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ISSN: 0030-8730
The equivalence of group invariant positive definite functions

Thomas Ru-Wen Chow

Vol. 26 (1968), No. 1, 25–38
Abstract

Let G be a separable locally compact group; ρ, a positive definite function; M(G), the set of all finite Radon measures; and

                          ∫    ∫
Np = {α ∈ M (G ) | B ρ(α,α ) ≡     ρ(t−1s)α (ds)α(dt) = 0}.
G×G

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 TT 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.

Mathematical Subject Classification
Primary: 46.80
Secondary: 22.00
Milestones
Received: 29 November 1966
Published: 1 July 1968
Authors
Thomas Ru-Wen Chow