Schoenberg's theorem for real and complex Hilbert spheres revisited

TL;DR

This paper extends Schoenberg's theorem to complex and real Hilbert spheres, characterizing positive definite functions via uniform expansions and limits from finite-dimensional spheres, with applications to group theory and harmonic analysis.

Contribution

It generalizes Schoenberg's theorem to complex and real Hilbert spheres, providing explicit expansions and limit processes for positive definite functions on these spheres.

Findings

Characterization of positive definite functions via uniform power series expansions.

Extension of Schoenberg's theorem to complex and real Hilbert spheres.

Derivation of coefficient functions as limits from finite-dimensional sphere expansions.

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.