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


Author: Yuan Xu
Journal: Proc. Amer. Math. Soc. 146 (2018), 2039-2048
MSC (2010): Primary 33C45, 33C50, 42A82, 60E10
DOI: https://doi.org/10.1090/proc/13913
Published electronically: October 30, 2017
MathSciNet review: 3767355
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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.


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Additional Information

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
MR Author ID: 227532
Email: yuan@uoregon.edu

Keywords: Positive definite functions, sphere, positive integrals, Jacobi polynomials
Received by editor(s): January 10, 2017
Received by editor(s) in revised form: July 1, 2017
Published electronically: October 30, 2017
Additional Notes: The author was supported in part by NSF Grant DMS-1510296.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2017 American Mathematical Society