Isotropic positive definite functions on spheres generated from those in Euclidean spaces

Abstract:

For a continuous function $g(x)$ on $[0, \infty )$ with $g(x) =0, x \ge \pi$, if it satisfies the inequality \begin{equation*} \int _0^\pi u^{\alpha +\frac {1}{2}} g(u) J_{\alpha -\frac {1}{2}} (x u) du \ge 0, ~~~~~ x \ge 0, \end{equation*} then it is shown in this paper that \begin{equation*} \int _0^\pi g(u) P_n^{ (\alpha )} (\cos u ) \sin ^{2 \alpha } u du \ge 0, ~~~~~~ n \in \mathbb {N}, \end{equation*} where $\alpha$ is a nonnegative integer, and $J_\nu (x)$ and $P_n^{ (\nu )} (x)$ denote the Bessel function and the ultraspherical polynomial, respectively. As a consequence, for an isotropic and continuous positive definite function in the Euclidean space, if it is compactly supported, it can be adopted as an isotropic positive definite function on a sphere.