Approximate Stable Diagonalization of the Green's Function for Low Frequencies

We present an approximate diagonalization of the Green's function that is stable at arbitrarily short distances with respect to wavelength. The diagonalization is based on scaled spherical functions and plane waves, where a scaling factor is used to stabilize special functions with small arguments. Optimization of the scaling factor leads to accurate diagonalizations, which can be used to implement the multilevel fast multipole algorithm for low-frequency problems.