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In this paper, we aim to derive some error bounds for Filon-ClenshawCurtis quadrature for highly oscillatory integrals. Thanks to the asymptotics of the coefficients in the Chebyshev series expansions of analytic functions or functions of limited regularities, these bounds are established by the aliasings of Fourier transforms on Chebyshev polynomials together with van der Corput-type lemmas. These errors share the property that the errors decrease with the increase of the frequency ω. Moreover, for fixed ω, the order of the error bound related to the number of interpolation nodes N is attainable, while for fixed N , the order of the error on ω is attainable too, which is verified by some functions of limited regularities. In particular, if the functions are analytic in Bernstein ellipses, then the errors decay exponentially. Furthermore, for large values of ω, the accuracy can be further improved by applying a special Hermite interpolants in the Filon-Clenshaw-Curtis quadrature, which can be efficiently evaluated by the Fast Fourier Transform (FFT) techniques.

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Based upon the fast computation of the coefficients of the interpolation polynomials at Chebyshev-type points by FFT, DCT and IDST, respectively, together with the efficient evaluation of the modified moments by forwards recursions or by the Oliver's algorithm, this paper presents interpolating integration algorithms, by using the coefficients and modified moments, for Clenshaw-Curtis, Fejér's first and second-type rules for Jacobi or Jacobi weights multiplied by a logarithmic function. The corresponding Matlab codes are included. Numerical examples illustrate the stability, accuracy of the Clenshaw-Curtis, Fejér's first and second rules, and show that the three quadratures have nearly the same convergence rates as Gauss-Jacobi quadrature for functions of finite regularities for Jacobi weights, and are more efficient upon the cpu time than the Gauss evaluated by fast computation of the weights and nodes by Chebfun.

In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods.