Schoenberg's theorem for real and complex Hilbert spheres revisited

Autor(es)

Berg, Christian ^{[1, 2]} ; Peron, Ana P. ^[1] ; Porcu, Emilio ^{[1, 3]}

Afiliação do(s) autor(es)

^[1] Univ Tecn Federico Santa Maria, Dept Math, Av Espana 1680, Valparaiso 2390123 - Chile

^[2] Univ Copenhagen, Dept Math Sci, Univ Pk 5, DK-2100 Copenhagen - Denmark

^[3] Newcastle Univ, Sch Math & Stat, Newcastle Upon Tyne, Tyne & Wear - England

• Tipo de documento: Artigo Científico
• Fonte: Journal of Approximation Theory; v. 228, p. 58-78, APR 2018.
• Citações Web of Science: 6

Resumo

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 ID) denote the closed unit disc in the complex plane. Continuous functions f : (D) over bar x L -> C such that f (xi . eta, u(-1) v) is a positive definite kernel on the product of the unit sphere in l(2)(C) and L are characterized as the functions with a uniformly convergent expansion f(z, u) = Sigma(m,n=0) (infinity) phi(m,n)(u)z(m-n)z, where phi(m,n) is a double sequence of continuous positive definite functions on L such that Sigma phi(m,n)(e(L)) < infinity (e(L) is the neutral element of L). It is shown how the coefficient functions phi(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. (C) 2018 Elsevier Inc. All rights reserved. (AU)

• Processo FAPESP: 16/03015-7 - Funções positivas definidas
• Beneficiário: Ana Paula Peron
• Modalidade de apoio: Bolsas no Exterior - Pesquisa
• Processo FAPESP: 14/25796-5 - Positividade (estrita) definida e diferenciabilidade
• Beneficiário: Ana Paula Peron
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 16/09906-0 - Análise harmônica, teoria da aproximação, funções especiais e aplicações
• Beneficiário: Dimitar Kolev Dimitrov
• Modalidade de apoio: Auxílio à Pesquisa - Temático