A close look at the entropy numbers of the unit ball of the Reproducing Hilbert Space of isotropic positive definite kernels

We present upper and lower bounds for the covering numbers, with accurate explicit constants, of the unit ball for two general classes of Reproducing Kernel Hilbert Space (RKHS) on the unit sphere of Rd+1. In both classes, the RKHS is generated by an isotropic continuous positive definite kernel and the bounds we present carry precise information about the asymptotic constants, depending on the dimension of the sphere and the monotonic behavior of the Schoenberg/Fourier coefficients of the isotropic kernel.