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S. Börm, J. Melenk:
"Approximation of the high-frequency Helmhotz kernel by nested directional interpolationn";
Numerische Mathematik, 137 (2017), 1; S. 1 - 34.

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 post-multiplication 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.

http://dx.doi.org/10.1007/s00211-017-0873-y