A P\'olya criterion for (strict) positive definiteness on the sphere

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, $f(\cos \cdot)$, 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 P\'olya 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.