  We present and analyze an approximation scheme for a class of highly
oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as
examples. The scheme is based on polynomial interpolation combined with
suitable pre- and postmultiplication by plane waves. It is shown to converge
exponentially in the polynomial degree and supports multilevel approximation
techniques. Our convergence analysis may be employed to establish exponential
convergence of certain classes of fast methods for discretizations of the
Helmholtz integral operator that feature polylogarithmic-linear complexity.
