Bounds on the error of fejer and clenshaw-curtis type quadrature for analytic functions

We consider the problem of integrating a function f:[-1, 1]->R which has an analytic extension f to an open disk Dr of radius r and center the origin, such that |f(z)| <= 1 for any z ? Dr. The goal of this paper is to study the minimal error among all algorithms which evaluate the integrand at the zeros of the n-degree Chebyshev polynomials of first or second kind (Fejer type quadrature formulas) or at the zeros of (n-2)-degree Chebyshev polynomials jointed with the endpoints -1,1 (Clenshaw-Curtis type quadrature formulas), and to compare this error to the minimal error among all algorithms which evaluate the integrands at n points. In the case r > 1, it is easy to prove that Fejer and Clenshaw-Curtis type quadrature are almost optimal. In the case r=1, we show that Fejer type formulas are not optimal since the error of any algorithm of this type is at least about n?-2. These results hold for both the worst-case and the asymptotic settings.