A Polya criterion for (strict) positive-definiteness on the sphere

Positive-definite functions are very important in both the theory and applications of approximation theory, probability and statistics. In particular, identifying strictly positive-definite kernels is of great interest as interpolation problems based upon such kernels are guaranteed to have a unique solution whenever the nodes {x(j)} are distinct. A Bochner-type result of Schoenberg characterizes continuous positive-definite zonal functions, f(cos center dot), on the sphere Sd-1, as those with non-negative Gegenbauer coefficients. More recent results characterize strictly positive-definite functions on Sd-1 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 Polya-type criterion for functions to be (strictly) positive definite on the sphere Sd-1. In analogy to the case of 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.