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International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 12, Pages 715-724
http://dx.doi.org/10.1155/S0161171202108076

Strict positive definiteness on spheres via disk polynomials

V. A. Menegatto1 and A. P. Peron2

1Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, São Carlos 13560-970, SP, Brazil
2Departamento de Matemática, CCE - Universidade Estadual de Maringá, Avenida Colombo 5790, Maringá 87020-900, PR, Brazil

Received 16 August 2001; Revised 3 April 2002

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We characterize complex strictly positive definite functions on spheres in two cases, the unit sphere of ℂq, q≥3, and the unit sphere of the complex ℓ2. The results depend upon the Fourier-like expansion of the functions in terms of disk polynomials and, among other things, they enlarge the classes of strictly positive definite functions on real spheres studied in many recent papers.