Abstract
Usually, the fast evaluation of a convolution integral \(\int_{{\mathbb{R}}}f(y)g(x-y)\mathrm{d}y\) requires that the functions f, g are discretised on an equidistant grid in order to apply the fast Fourier transform. Here we discuss the efficient performance of the convolution in locally refined grids. More precisely, the convolution result is projected into some given locally refined grid (Galerkin approximation). Under certain conditions, the overall costs are still \({\mathcal{O}}(N\log N),\) where N is the sum of the dimensions of the subspaces containing f, g and the resulting function.
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Hackbusch, W. Fast and exact projected convolution for non-equidistant grids. Computing 80, 137–168 (2007). https://doi.org/10.1007/s00607-007-0229-z
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DOI: https://doi.org/10.1007/s00607-007-0229-z