Fast expansion into harmonics on the ball
<|reference_start|>Fast expansion into harmonics on the ball: 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 their dense counterparts have time complexity $O(N^6)$. We illustrate our methods on numerical examples.<|reference_end|>
@article{kileel2024fast,
    title={Fast expansion into harmonics on the ball},
    author={Joe Kileel, Nicholas F. Marshall, Oscar Mickelin, Amit Singer},
    journal={arXiv preprint arXiv:2406.05922},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.05922},
    primaryClass={math.NA cs.NA math.CA}
}