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MiS Preprint

Fast and Exact Projected Convolution of Piecewise Linear Functions on Non-equidistant Grids - Extended Version

Wolfgang Hackbusch


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, $f$ and $g$ are assumed to be piecewise linear and the convolution result is projected into the space of linear functions in a given locally refined grid. 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.

MSC Codes:
44A35, 42A55
convolution integral

Related publications

2008 Repository Open Access
Wolfgang Hackbusch

Fast projected convolution of piecewise linear functions on non-equidistant grids

In: From nano to space / Michael H. Breitner (ed.)
Berlin : Springer, 2008. - pp. 145-160