Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals

In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form integral(1)(-1) f(s) exp(iks) ds. We use a Filon approach based on interpolating f at the classical Clenshaw-Curtis points cos(j pi/N), j = 0, ... , N. The rule may be implemented in O(N log N) operations. We prove error estimates that show explicitly how the error depends both on the parameters k and N and on the Sobolev regularity of f. In particular we identify the regularity of f required to ensure the maximum rate of decay of the error as k -> infinity. We also describe a method for implementing the method and prove its stability both when N <= k and N > k. Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates.