Kernels of many important convolution operators admit an accurate and efficient approximation by a linear combination of Gaussians. Such approximation yields their separated representation and, as a result, a fast algorithm for applying these operators; an approach already used in quantum chemistry.
It turns out that kernels of a number of non-convolution operators also have accurate and efficient approximations via Gaussians. We will discuss two examples, namely, Green's functions satisfying boundary conditions on simple domains and the Green's function for the harmonic oscillator in quantum mechanics. These examples lead to a problem of approximating functions of several variables via exponentials as a way to extend the approach to a larger class of operators.
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