Positive definite functions on the unit sphere and integrals of Jacobi polynomials
Abstract
It is shown that the integrals of the Jacobi polynomials \begin{equation*}%\label{eq:Fn^J} \int_0^t (t-\theta)^\delta P_n^{(\alpha-\frac12,\beta-\frac12)}(\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 \cite{BCX} 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 Polyà criterion for positive definite functions on the sphere for all dimensions. Moreover, the positive definiteness of the function $\theta \mapsto (t - \theta)_+^\delta$ is also established on the compact two-point homogeneous spaces.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2017
- DOI:
- 10.48550/arXiv.1701.00787
- arXiv:
- arXiv:1701.00787
- Bibcode:
- 2017arXiv170100787X
- Keywords:
-
- Mathematics - Classical Analysis and ODEs;
- 33C45;
- 33C50;
- 42A82;
- 60E10
- E-Print:
- Final form. Accepted Proc. AMS