Error bounds for approximation in Chebyshev points
\textit{Lloyd N. Trefethen} [SIAM Rev. 50, No.~1, 67--87 (2008; Zbl 1141.65018)] has compared the convergence behavior of the Gauss quadrature with the Clenshaw-Curtis quadrature [cf. \textit{C. W. Clenshaw} and \textit{A. R. Curtis}, Numer. Math. 2, 197--205 (1960; Zbl 0093.14006)] and the experiments show that the supposed factor-of-2 advantage of the Gauss quadrature is rarely realized and backed by the corresponding theorems explaining this effect. These results are employed in this paper to consider new error estimates for approximations of \(f\) in the Chebyshev points. It is demonstrated that polynomial interpolation in the Chebyshev points of the 1st and 2nd kind should be regarded as equally valuable and fundamental. Error bounds for Gauss, Clenshaw-Curtis and Fejér's first quadratures are improved by using new error estimates for polynomial interpolation in the Chebyshev points. Numerical results (for highly oscillatory integrals) demonstrate that the improved error bounds are reasonably sharp. These results can be employed for approximate solutions of integral equations appearing in identification and control theory of nonlinear dynamical systems [\textit{D. N. Sidorov}, Sib. Zh. Ind. Mat. 3, No.~1, 182--194 (2000; Zbl 0951.93021)].
scientific article; zbMATH DE number 5797349
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