2011/10/11 by R. K. Beatson, Beatson, R. K., Wolfgang zu Castell +3 · 1 citation
Engineering · Mathematics · #41A05 #41A63 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.1110.2437
openalex publication_date 2011/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Positive definite functions are very important in both theory and applications of approximation theory, probability and statistics. In particular, identifying strictly positive definite kernels is of great interest as interpolation problems corresponding to these kernels are guaranteed to be poised. A Bochner type result of Schoenberg characterises continuous positive definite zonal functions, f(cos ⋅), on the sphere \Sdmone, as those with nonnegative Gegenbauer coefficients. More recent results characterise strictly positive definite functions on \Sdmone by stronger conditions on the signs of the Gegenbauer coefficients. Unfortunately, given a function f, checking the signs of all the Gegenbauer coefficients can be an onerous, or impossible, task. Therefore, it is natural to seek simpler sufficient conditions which guarantee (strict) positive definiteness. We state a conjecture which leads to a Pólya type criterion for functions to be (strictly) positive definite on the sphere \Sdmone. In analogy to the case of the Euclidean space, the conjecture claims positivity of a certain integral involving Gegenbauer polynomials. We provide a proof of the conjecture for d from 3 to 8.