Optimal quadrature problem on Hardy-Sobolev classes

Let denote those 2 pi-periodic, real-valued functions f on R, which are analytic in the strip S-beta, := (z is an element of C : vertical bar Imz vertical bar < beta}, beta > 0 and satisfy the restriction vertical bar f((r))(z)vertical bar <= 1, z is an element of S-beta. Denote by [x] the integral part of x. We prove that the rectangular formula N-I 27ri Q*N(f) N j=0 is optimal for the class of functions H among all quadrature formulae of the form QN*(f) = 2 pi/N Sigma(j=0)(N-1) f(2 pi j/N) where the nodes 0,<= t(1) < ... < t(n) < 27r and the coefficients aij E R are arbitrary, i = 1,... 1 17, j = 0, 1, vi - 1, and (vI, vn) is a system of positive integers satisfying the condition Sigma(i=1)(n)2[(v(i) + 1)/2 <= 2N. In particular, the rectangular formula is optimal for the class of functions H-infinity,beta(r) among all quadrature formulae of the form: QN(f) = Sigma(i=1)(N) a(i)f(t(i)), where O <= t(1) < ... < t(N) < 2 pi and a(i) is an element of R, i = 1, .... N. Moreover, we exactly determine the error estimate of the optimal quadrature formulae on the class (H) over tilde infinity,beta. (c) 2005 Elsevier Inc. All rights reserved.