Covering numbers for real-valued function classes

As a byproduct of Theorem 3.1 we can give an estimate of the truncation error which arises if one ignores all the samples outside a finite interval. More precisely, we have the following corollary. Theorem 4.1: Let us define the truncation error E N (t) as follows: E N (t) = F(t) 0 jnjN F (t n)S n (t): Then jE N (t)j 2 2p0q bA p C ^ (1 0 p D)(p 0 q 0 1)N p0q01 ; t in the compact set K (14) where p > q + 1, and q the polynomial order of growth of F (t): Proof: We know from Theorem 3.1 that jF(tn)j bjtnj q for some q 0 and a constant b, and jS n (t)j 2 p A p C ^ (1 0 p D)jtnj p = CK jtnj p ; for all p 0 and t in the compact set K: Hence, if we take any p > q + 1, we will have jE N (t)j jnj>N jF(t n)j jS n (t)j bCK jnj>N