Berg C

Peron AP

Porcu E

\ua9 2018 Elsevier Inc. Schoenberg\u27s 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 D\uaf denote the closed unit disc in the complex plane. Continuous functions f:D\uaf
7L→C such that f(ξ⋅η,u−1v) is a positive definite kernel on the product of the unit sphere in ℓ2(C) and L are characterized as the functions with a uniformly convergent expansion f(z,u)=∑m,n=0∞φm,n(u)zmz\uafn,where φm,n is a double sequence of continuous positive definite functions on L such that ∑φm,n(eL)&lt;∞ (eL is the neutral element of L). It is shown how the coefficient functions φm,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

Schoenberg\u27s theorem for real and complex Hilbert spheres revisited

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