We show how a sparse discretisation matrix can be obtained for an integral operator Ak with an oscillatory kernel function Gk(x, y) = G1(x, y) exp(ikG2(x,y)), where k is a parameter that explicitly determines the frequency of this function. The first result is that, for a fixed function u(y), the function (Aku) (x) can be computed efficiently and with arbitrary accuracy, even as k grows very large. The number of operations is in fact bounded in k. The method can be extended to evaluate (Akuk) (x), where uk(y) = u1(y) exp(iku2(y)) may be increasingly oscillatory.
We proceed to the solution of oscillatory integral equations of the form (λI+Ak) u = fk. Such problems are considerablymore complicated than the problem of applying an integral operator. We show that a sparse discretisation remains possible in the case of scattering by a convex obstacle, leading immediately to an efficient solution method.