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Eigenvalue decay of positive integral operators on the sphere

Authors: M. H. Castro and V. A. Menegatto
Journal: Math. Comp. 81 (2012), 2303-2317
MSC (2010): Primary 45C05, 47B38, 47G10, 42A82, 47A75, 45P05, 41A36, 45M05
Published electronically: March 20, 2012
MathSciNet review: 2945157
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Abstract: We obtain decay rates for singular values and eigenvalues of integral operators generated by square integrable kernels on the unit sphere in $ \mathbb{R}^{m+1}$, $ m\geq 2$, under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined using standard differentiation in $ \mathbb{R}^{m+1}$. In this paper, the assumptions are all defined via the Laplace-Beltrami derivative, a concept first investigated by Rudin in the early fifties and genuinely spherical in nature. The rates we present depend on both, the differentiability order used to define the smoothness conditions and the dimension $ m$. They are shown to be optimal.

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Additional Information

M. H. Castro
Affiliation: Faculdade de Matemática, Universidade Federal de Uberlândia, Caixa Postal 593, 38400-902, Uberlândia-MG, Brasil

V. A. Menegatto
Affiliation: Departamento de Matemática, ICMC-USP, São Carlos, Caixa Postal 668, 13560-970, São Carlos-SP, Brasil

Keywords: Sphere, integral operators, eigenvalues, singular values, eigenvalue estimates, positive definite kernels, Laplace-Beltrami differentiability, Laplace-Beltrami integral.
Received by editor(s): January 12, 2011
Received by editor(s) in revised form: June 22, 2011
Published electronically: March 20, 2012
Additional Notes: The first author was partially supported by CAPES-Brasil.
The second author was partially supported by FAPESP, Grant #2010/19734-6
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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