Schoenberg's theorem for real and complex Hilbert spheres revisited

Abstract

Schoenberg's theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let \overline{\D} denote the closed unit disc in the complex plane. Continuous functions f:\overline{\D}\times L\to \C such that f(\xi \cdot \eta,u^{-1}v) is a positive definite kernel on the product of the unit sphere in \ell_2(\C) and L are characterized as the functions with a uniformly convergent expansion f(z,u)=\sum_{m,n=0}^\infty \varphi_{m,n}(u)z^m\overline{z}^n, where \varphi_{m,n} is a double sequence of continuous positive definite functions on L such that \sum\varphi_{m,n}(e_L)<\infty (e_L is the neutral element of L). It is shown how the coefficient functions \varphi_{m,n} are obtained as limits from expansions for positive definite functions on finite dimensional complex spheres via a Rodrigues formula for disc polynomials. Similar results are obtained for the real Hilbert sphere.