Capacity of reproducing kernel spaces in learning theory

The capacity of reproducing kernel Hilbert spaces (RKHS) plays an essential role in the analysis of learning theory. Covering numbers and packing numbers of balls of these reproducing kernel spaces are important measurements of this capacity. In this paper, we first present lower bound estimates for the packing numbers by means of nodal functions. Then we show that if a Mercer kernel is C^s (for some s > 0 being not an even integer), the RKHS associated with this kernel can be embedded into C^s/2. This gives upper-bound estimates for the covering number concerning Sobolev smooth kernels. Examples and applications to V_γ dimension and Tikhonov regularization are presented to illustrate the upper- and lower-bound estimates.