Istituto di Scienza e Tecnologie dell'Informazione     
Favati P., Lotti G., Romani F. Bounds on the error of fejer and clenshaw-curtis type quadrature for analytic functions. In: Applied Mathematics Letters, vol. 6 (6) pp. 3 - 8. Pergamon, 1993.
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.

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