The covering number for some Mercer kernel Hilbert spaces

In the present paper, we investigate the estimates for the covering number of a ball in a Mercer kernel Hilbert space on [0,1][0,1]. Let Pl(x)Pl(x) be the Legendre orthogonal polynomial of order l  , al>0al>0 be real numbers satisfying ∑l=0+∞lal<+∞. Then, for the Mercer kernel functionK(x,t)=∑l=0+∞alPl(x)Pl(t),x,t∈[0,1],we provide the upper estimates of the covering number for the Mercer kernel Hilbert space reproducing from K(x,t)K(x,t). For some particular alal we give the lower estimates. Meanwhile, a kind of l2l2-norm estimate for the inverse Mercer matrix associated with the Mercer kernel K(x,t)K(x,t) is given.