Positive definite kernels on the complex Hilbert sphere
type: Item
instance of: scholarly article
title: Positive definite kernels on the complex Hilbert sphere
zbMATH Open document ID: 0572.43004
DOI: 10.1007/BF01318903
author: J. P. R. Christensen
author: Paul Ressel
published in: Mathematische Zeitschrift
publication date: 1982-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173180
review text: Let H be an infinite-dimensional complex Hilbert space, and let L be the set of complex-valued functions f on \(D:=\{z\in {\mathbb{C}}:\) \(| z| \leq 1\}\) such that \((x,y)\mapsto f(<x,y>)\) is a positive definite kernel on the unit sphere S of H. The authors prove that a continuous function \(f\in L\) has a unique series representation \(f(z)=\sum a_{n,m} z^ n\bar z^ m\) (z\(\in D)\), where all \(a_{n,m}\geq 0\) and \(\sum a_{n,m}<\infty\). Following a hint of I. J. Schoenberg they identify \(L_ 1:=\{f\in L:\) \(f(1)=1\}\) with the set K of all functions \(f: D\to {\mathbb{C}}\) with \(f(1)=1\) and the following property: If \((a_{jk})\) is a positive semidefinite complex matrix then \((f(a_{jk}))\) is positive semidefinite. They show that K is a Bauer simplex whose set of extreme points is the closure of \(\{z^ n\bar z^ m:\) \(n,m\in {\mathbb{N}}_ 0\}\) in \({\mathbb{C}}^ D\), and establish their result by using an approximation argument. The real analogue was shown by Schoenberg in 1942, and the authors have given a new proof of Schoenberg's result by the method just described [Trans. Am. Math. Soc. 243, 89-95 (1978; Zbl 0394.15011)]. For similar results of C. S. Herz, D. Rider, C. C. Graham and W. Moran, replacing positive semidefinite matrices by positive definite functions on infinite abelian groups, see the book by \textit{C. C. Graham} and \textit{O. C. McGehee} [Essays in commutative harmonic analysis (1979; Zbl 0439.43001), Theorem 9.6.1].
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 46E20
zbMATH DE Number: 3912978
zbMATH Keywords: complex Hilbert space
zbMATH Keywords: positive definite kernel
zbMATH Keywords: unique series representation
zbMATH Keywords: Bauer simplex
MaRDI profile type: Publication
cites work: Functions Operating on Positive Definite Matrices and a Theorem of Schoenberg
cites work: Q3881310
cites work: Positive-definite functions on involution semigroups
cites work: Positive definite functions on spheres
OpenAlex ID: W2085854214
description: scientific article; zbMATH DE number 3912978