Fast expansion into harmonics on the ball

• DOI: 10.48550/arxiv.2406.05922
• Publication Date: 2024-06-09
• Authors: Joe Kileel; Nicholas F. Marshall; Oscar Mickelin; Amit Singer
• Keywords: Ball (mathematics)

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