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arXiv · 2406.05922

Fast expansion into harmonics on the ball

Joe Kileel·Nicholas F. Marshall·Oscar Mickelin·Amit Singer
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Abstract

We devise fast and provably accurate algorithms to transform between an $N\times N \times N$ Cartesian voxel representation of a three-dimensional function and its expansion into the {ball harmonics}, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in $\mathbb{R}^3$. Given $\varepsilon > 0$, our algorithms achieve relative $\ell^1$ - $\ell^\infty$ accuracy $\varepsilon$ in time $O(N^3 (\log N)^2 + N^3 |\log \varepsilon|^2)$, while the na\"{i}ve direct application of the expansion operators has time complexity $O(N^6)$. We illustrate our methods on numerical examples.

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Joe Kileel, Nicholas F. Marshall, Oscar Mickelin, Amit Singer. 2024-06-09. Fast expansion into harmonics on the ball. https://doi.org/10.1137/24m1668159

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