Eigenvalue decay of positive integral operators on the sphere

Castro, M.; Menegatto, V.

doi:10.1090/S0025-5718-2012-02595-6

Eigenvalue decay of positive integral operators on the sphere
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by M. H. Castro and V. A. Menegatto PDF

Math. Comp. 81 (2012), 2303-2317 Request permission

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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