Strict positive definiteness on spheres via disk polynomials

V A Menegatto,A P Peron

doi:10.1155/s0161171202108076

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.

Highlights

This paper is concerned with positive definite kernels that are useful to perform interpolation on spheres
We will provide an elementary description of strict positive definiteness of kernels which are inner-product dependent, that is, of the form (z, w) ∈ Ω2q × Ω2q → f z, w, (1.1)
We collect basic results about positive definite functions on spheres and disk polynomials. This is crucial in the paper, because the analysis of strict positive definiteness is based on the expansion of positive definite functions on Ω2q in terms of certain disk polynomials

Summary

Introduction

This paper is concerned with positive definite kernels that are useful to perform interpolation on spheres. ΛN , and a strictly positive definite function f of order at least N on Ω2q, there is a unique function of the form The results in this paper will provide a concise way of constructing complex strictly positive definite functions on the real spheres.

Results

Conclusion