  We present estimates for the covering numbers of the unit ball of Reproducing
Kernel Hilbert Spaces (RKHSs) of functions on $M^d$ a d-dimensional compact
two-point homogeneous space. The RKHS is generated by a continuous
zonal/isotropic positive definite kernel. We employ the representation in terms
of the Schoenberg/Fourier series expansion for continuous isotropic positive
definite kernels, given in terms of a family of orthogonal polynomials on
$M^d$. The bounds we present carry accurate information about the asymptotic
constants depending on the dimension of the manifold and the decay or growth
rate of the coefficients of the kernel. The results we present extend the
estimates previously known for continuous isotropic positive definite kernels
on the d-dimensional unit sphere. We present the weak asymptotic equivalence
for the order of the growth of covering numbers associated to kernels on $M^d$
with a convergent geometric sequence of coefficients. We apply our estimates in
order to present a bound for the covering numbers of the spherical Gaussian
kernel, and to present bounds for formal examples on $M^d$.
