On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals (Q2355180)

The authors continue their results, ref.: \textit{S. Xiang} et al. [Numer. Math. 116, No. 3, 463--491 (2010; Zbl 1201.65040)], where results of \textit{L. N. Trefethen} [SIAM Rev. 50, No. 1, 67--87 (2008; Zbl 1141.65018)] and \textit{C. W. Clenshaw} and \textit{A. R. Curtis} [Numer. Math. 2, 197--205 (1960; Zbl 0093.14006)] are used to consider new error estimates for approximations in the Chebyshev points. Using to the asymptotics of the coefficients in the Chebyshev series expansions of analytic functions or functions of limited regularities the error bounds for Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals are derived. Namely, the problem of the computation of \(I_{\omega} = \int_{-1}^1 f(x) e^{i\omega x}\, dx\) is analysed. Apart from suitably smooth functions \(f(x)\) error bounds are considered for integrands containing algebraical or logarithmical singularities. Theoretically justified convergence rates are tested using series of numerical experiments.