# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1701.07214