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.

翻译：我们设计了快速且可证明精确的算法，用于实现三维函数在$N\times N \times N$笛卡尔体素表示与球谐函数展开之间的转换，其中球谐函数是$\mathbb{R}^3$中单位球上狄利克雷拉普拉斯算子的特征基。给定精度参数$\varepsilon > 0$，我们的算法在$O(N^3 (\log N)^2 + N^3 |\log \varepsilon|^2)$时间内达到相对$\ell^1$ - $\ell^\infty$精度$\varepsilon$，而传统稠密算法的复杂度为$O(N^6)$。我们通过数值算例展示了所提方法的有效性。