Strictly positive definite functions on a compact group

Abstract

We recognize a result of Schreiner, concerning strictly positive definite functions on a sphere in an Euclidean space, as a generalization of Bochner's theorem for compact groups. The purpose of this note is to provide a rapid proof of a relatively new result of Schreiner, [S], concerning strictly positive definite functions on a sphere in a Euclidean space. It is our pleasure to see that the theory of such functions has found applications in geosciences, and that it is well rooted in the existing mathematical literature. Let G be a compact group, and let H C G be a closed subgroup, such that the quotient G/H is infinite. Let f: G C be a continuous function, invariant under the left and right translations by elements of H. We denote the space of all such functions by C(H\G/H). Following Schreiner, [S], we say that the function f is strictly positive definite if and only if n (1) E cicjf(xix j) > 0 i,j~=l for any finite set {Xl I X2, ..., Xn2} C G such that the cosets x1H, x2H, ..., xnH C G/H are distinct, and any complex numbers c1, c2, ..., cn, not all equal to zero. Let G denote the unitary dual of G, [K, 7.3]. This is the set of equivalence classes of irreducible unitary representations of G. For convenience, we choose an irreducible representation for each such class, and identify G with the set of these representations. Recall the Fourier transform (2) f(ir) =ir(f) = ir(x)f(x) dx (e CZ) where f is any absolutely integrable function on G with respect to the Haar measure dx. Thus each ir(f) is a linear map on the finite dimensional Hilbert space 'H1,, where the representation 7r is realized. Let 'H = {v E R, : ir(h)v = v, h E H} be the subspace of H-fixed vectors. Let (G/H) = {r E C; THH A 0}. By the Robenius reciprocity theorem, (G/H)^ is the subset of G, consisting of representations which occur in L2(G/H) (see [K, 8.4]). Received by the editors December 18, 1998 and, in revised form, August 30, 1999. 1991 Mathematics Subject Classification. Primary 43A35, 43A90, 42A82, 41A05.