A Pólya 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 {xj} are distinct. A Bochner-type result of Schoenberg characterizes continuous positive-definite zonal functions, f(cos ·), on the sphere ^d−1, as those with non-negative Gegenbauer coefficients. More recent results characterize strictly positive-definite functions on ^d−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 Pólya-type criterion for functions to be (strictly) positive definite on the sphere ^d−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.