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We present formulas for accurate numerical conversion between functionsrepresented by multiwavelets and their multipole/local expansions with respectto the kernel of the form, $e^{\lambda r}/r$. The conversion is essential forthe application of fast multipole methods for functions represented bymultiwavelets. The corresponding separated kernels exhibit near-singularbehaviors at large $\lambda$. Moreover, a multiwavelet basis functionoscillates more wildly as its degree increases. These characteristics incombination render any brute-force approach based on numerical quadraturesimpractical. Our approach utilizes the series expansions of the modifiedspherical Bessel functions and the Cartesian expansions of solid harmonics sothat the multipole-multiwavelet conversion matrix can be evaluated like aspecial function. The result is a quadrature-free, fast, reliable, and machineprecision accurate scheme to compute the conversion matrix with predictablesparsity patterns.

An accurate and quadrature-free evaluation of multipole expansion of functions represented by multiwavelets