TABLE OF CONTENTS First we get a lower bound for the error of formula (A.3) with ? 1 = ? 2 = 0 and n 1 = n 2 = n, on H lder classes.
Let ? = H ??( D) , and calculate the integral Tf by formula (A.3) with n 1 =n 2 =n and ? 1 = ? 2 = 0. Then the estimate
| (A.28) |  |
holds.
Proof. Let n > 0 be an integer, L = [ n/log n]. Let v k := -1 + 2 k/ L, k = 0, 1, ..., L. By ( ? k, ? l) we denote a set which is the union of nodes ( x i, y j) , i,j = 1, 2, ..., n of formula (A.3) and the nodes ( v i,v j) , i,j = 1, 2, ..., L. Let ? kl = [ v k,v k +1 ;v l,v l +1] , k, l = 0, 1, ..., L - 1. Let 0 ? ?( t 1 ,t 2) ? H ??( D), where D = [-1, 1] 2, vanishing at the nodes ( ? k,
