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


Authors: Zhihui Nie and Chunsheng Ma
Journal: Proc. Amer. Math. Soc. 147 (2019), 3047-3056
MSC (2010): Primary 33C45, 33C50, 42A82, 60E10
DOI: https://doi.org/10.1090/proc/14454
Published electronically: April 3, 2019
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Abstract: For a continuous function $ g(x)$ on $ [0, \infty )$ with $ g(x) =0, x \ge \pi $, if it satisfies the inequality

$\displaystyle \int _0^\pi u^{\alpha +\frac {1}{2}} g(u) J_{\alpha -\frac {1}{2}} (x u) du \ge 0, ~~~~~ x \ge 0,$    

then it is shown in this paper that

$\displaystyle \int _0^\pi g(u) P_n^{ (\alpha )} (\cos u ) \sin ^{2 \alpha } u du \ge 0, ~~~~~~ n \in \mathbb{N},$    

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.

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Additional Information

Zhihui Nie
Affiliation: Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, Kansas 67260-0033
Email: niezhihui2014@gmail.com

Chunsheng Ma
Affiliation: Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, Kansas 67260-0033
Email: chunsheng.ma@wichita.edu

DOI: https://doi.org/10.1090/proc/14454
Keywords: Positive definite function, compactly support function, sphere, Bessel function, Jacobi polynomial, ultraspherical polynomial
Received by editor(s): September 19, 2017
Received by editor(s) in revised form: August 1, 2018, and October 15, 2018
Published electronically: April 3, 2019
Communicated by: Yuan Xu
Article copyright: © Copyright 2019 American Mathematical Society