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Schoenberg's theorem for real and complex Hilbert spheres revisited

Lookup NU author(s): Professor Emilio Porcu

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Abstract

© 2018 Elsevier Inc. 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 D¯ denote the closed unit disc in the complex plane. Continuous functions f:D¯×L→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¯n,where φm,n is a double sequence of continuous positive definite functions on L such that ∑φm,n(eL)<∞ (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.


Publication metadata

Author(s): Berg C, Peron AP, Porcu E

Publication type: Article

Publication status: Published

Journal: Journal of Approximation Theory

Year: 2018

Volume: 228

Pages: 58-78

Print publication date: 01/04/2018

Online publication date: 07/02/2018

Acceptance date: 02/02/2018

ISSN (print): 0021-9045

ISSN (electronic): 1096-0430

Publisher: Academic Press Inc.

URL: https://doi.org/10.1016/j.jat.2018.02.003

DOI: 10.1016/j.jat.2018.02.003


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