Asymptotic Analysis: Entropy Numbers and Schoenberg Coefficients of Isotropic Positive Definite Reproducing Kernels

The harmonic analysis on the sphere is considered the basic setting in order to present precise constants in the bounds for the covering numbers of the unit ball of the Reproducing Hilbert Space of isotropic positive definite kernels. The bounds present information on the dimension contained in the growth assumptions on the coefficients of the isotropic kernel and permit us to respond to the important question about the existence of the asymptotic constant in this context. The basic decay of the Schoenberg coefficients of the kernels satisfying the assumptions of the main results is presented for odd dimensions.