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Positive definite functions are very important in both theory andapplications of approximation theory, probability and statistics. Inparticular, identifying strictly positive definite kernels is of great interestas interpolation problems corresponding to these kernels are guaranteed to bepoised. A Bochner type result of Schoenberg characterises continuous positivedefinite zonal functions, $f(\cos \cdot)$, on the sphere $\Sdmone$, as thosewith nonnegative Gegenbauer coefficients. More recent results characterisestrictly positive definite functions on $\Sdmone$ by stronger conditions on thesigns of the Gegenbauer coefficients. Unfortunately, given a function $f$,checking the signs of all the Gegenbauer coefficients can be an onerous, orimpossible, task. Therefore, it is natural to seek simpler sufficientconditions which guarantee (strict) positive definiteness. We state aconjecture which leads to a P\'olya type criterion for functions to be(strictly) positive definite on the sphere $\Sdmone$. In analogy to the case ofthe Euclidean space, the conjecture claims positivity of a certain integralinvolving Gegenbauer polynomials. We provide a proof of the conjecture for $d$from 3 to 8.

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