  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.

Classical Analysis and ODEs