Positive definite functions on the unit sphere and integrals of Jacobi polynomials

Abstract:

For $\alpha , \beta \in \mathbb {N}_0$ and $\max \{\alpha ,\beta \} >0$, it is shown that the integrals of the Jacobi polynomials \begin{equation*}\int _0^t (t-\theta )^\delta P_n^{(\alpha -\frac 12,\beta -\frac 12)}(\cos \theta ) \left (\sin \tfrac {\theta }2\right )^{2 \alpha } \left (\cos \tfrac {\theta }2\right )^{2 \beta } d\theta > 0 \end{equation*} for all $t \in (0,\pi ]$ and $n \in \mathbb {N}$ if $\delta \ge \alpha + 1$ for $\alpha ,\beta \in \mathbb {N}_0$ and $\max \{\alpha ,\beta \} > 0$. This proves a conjecture on the integral of the Gegenbauer polynomials in a work of Beatson (2014) that implies the strictly positive definiteness of the function $\theta \mapsto (t - \theta )_+^\delta$ on the unit sphere $\mathbb {S}^{d-1}$ for $\delta \ge \lceil \frac {d}{2}\rceil$ and the Pólya criterion for positive definite functions on the sphere $\mathbb {S}^{d-1}$ for $d \ge 3$. Moreover, the positive definiteness of the function $\theta \mapsto (t - \theta )_+^\delta$ is also established on the compact two-point homogeneous spaces.