Strict positive definiteness on spheres via disk polynomials
type: Item
instance of: scholarly article
title: Strict positive definiteness on spheres via disk polynomials
zbMATH Open document ID: 1039.42007
DOI: 10.1155/S0161171202108076
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-10-29T00:00:00Z
full work available at URL: https://eudml.org/doc/50792
review text: The paper under review is concerned with zonal spherical harmonic analysis. It studies kernels on the unit sphere \(\Omega_q\) of \({\mathbb C}^q\) admitting the form \(K_f: (z,w) \to f(\langle z| w\rangle)\) where the complex-valued function \(f\) is continuous on the closed unit disk of the complex plane \({\mathbb C}\), and \(\langle.| .\rangle\) denotes the standard scalar product of the complex vector space \({\mathbb C}^q, q \geq 3\). The function \(f\) is called the shape function of \(K_f\). Specifically, the problem of characterizing kernels \(K_f\) of strict positive type is considered. The characterization obtained depends on the expansion of the shape function \(f\) in terms of suitable disk polynomials. Disk polynomials can be expressed in terms of classical Jacobi polynomials [\textit{J. N. Boyd} and \textit{P. N. Raychowdhury}, ``Zonal harmonic functions from two-dimensional analogs of Jacobi polynomials'', Appl. Anal. 16, 243--259 (1983; Zbl 0503.33009)]. The main result of the paper demonstrates that the property of \(K_f\) of being a kernel of strict positive type can be read off from the index parity \(m - n\) of strictly positive coefficients \(a_{m,n}(f), m\geq 0, n\geq 0\), derived from the expansion of \(f\) in terms of disk polynomials [\textit{V. A. Menegatto} and \textit{A. P. Peron}, ``A complex approach to strict positive definiteness on spheres'', Integral Transform., Spec. Funct. 11, 377--396 (2001; Zbl 1025.42006); ``Positive definite kernels on complex spheres'', J. Math. Anal. Appl. 254, 219--232 (2001; Zbl 0973.32004)]. An extension of the main result to the unit sphere \(\Omega_\infty\) of \(\ell^2\) is also described.
Mathematics Subject Classification ID: 42A82
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 32A30
zbMATH DE Number: 1821314
zbMATH Keywords: zonal spherical harmonic analysis
zbMATH Keywords: disk polynomials
zbMATH Keywords: strict positive definiteness
author: Valdir A. Menegatto
author: Ana P. Peron
reviewed by: Walter Schempp
MaRDI profile type: Publication
OpenAlex ID: W2162344985
description: scientific article; zbMATH DE number 1821314