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 ∫ . We use a Filon approach based on interpolating f at the classical Clenshaw-Curtis points c . The rule may be implemented in O operations. We prove error estimates that show explicitly how the error depends both on the parameters k and N and on the Sobolev regularityof f. In particular we identify the regularity of f required to ensure the maximum rate of decay of the error as k → ∞. 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.