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(ξ\cdot η,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 φ_{m,n}(u)z^m\overline{z}^n, where φ_{m,n} is a double sequence of continuous positive definite functions on L such that \sumφ_{m,n}(e_L)<\infty (e_L is the neutral element of L). It is shown how the coefficient functions φ_{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.

22 pages

References in corpus (2)

• Orthogonal expansions related to compact Gelfand pairs
• Positive Definite Functions on Complex Spheres and their Walks through Dimensions

Cited by in corpus (7)

• A panorama of positivity
• Relations between Schoenberg Coefficients on Real and Complex Spheres of Different Dimensions
• Strictly positive definite kernels on the $2$-sphere: beyond radial symmetry
• Characterization of Strict Positive Definiteness on products of complex spheres
• A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere
• A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces
• Dimension walks on ${\mathbb{S}}^d\times \mathbb{R}$