Generalized Zernike or disc polynomials
Journal of Computational and Applied Mathematics
Model Selection for Regularized Least-Squares Algorithm in Learning Theory
Foundations of Computational Mathematics
Integral operators on the sphere generated by positive definite smooth kernels
Journal of Complexity
Radial basis functions and corresponding zonal series expansions on the sphere
Journal of Approximation Theory
Mercer’s theorem, feature maps, and smoothing
COLT'06 Proceedings of the 19th annual conference on Learning Theory
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We obtain explicit formulas for the eigenvalues of integral operators generated by continuous dot product kernels defined on the sphere via the usual gamma function. Using them, we present both, a procedure to describe sharp bounds for the eigenvalues and their asymptotic behavior near 0. We illustrate our results with examples, among them the integral operator generated by a Gaussian kernel. Finally, we sketch complex versions of our results to cover the cases when the sphere sits in a Hermitian space.