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Discretizing spherical integrals and its applications

Pages: 499 - 514, Issue special, November 2013

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Shaobo Lin - Institute for Information and System Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China (email)
Xingping Sun - Department of Mathematics, Missouri State University, Spring eld, MO 65810, United States (email)
Zongben Xu - Institute for Information and System Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China (email)

Abstract: Efficient discretization of spherical integrals is required in many numerical methods associated with solving differential and integral equations on spherical domains. In this paper, we discuss a discretization method that works particularly well with convolutions of spherical integrals. We utilize this method to construct spherical basis function networks, which are subsequently employed to approximate the solutions of a variety of differential and integral equations on spherical domains. We show that, to a large extend, the approximation errors depend only on the smoothness of the spherical basis function. We also derive error estimates of the pertinent approximation schemes. As an application, we discuss a Galerkin type solutions for spherical Fredholm integral equations of the first kind, and obtain rates of convergence of the spherical basis function networks to the solutions of these equations.

Keywords:  Spherical basis function, spherical convolution, approximation, Fredholm integral equation, sphere.
Mathematics Subject Classification:  Primary: 41A25, 41A05; Secondary: 41A63.

Received: October 2012;      Revised: February 2013;      Published: November 2013.

 References