{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/fast-expansion-into-harmonics-on-the-ball","title":"Fast expansion into harmonics on the ball","arxiv_id":"2406.05922","date":"2024-06-09","proceeding":null,"authors":["Joe Kileel","Nicholas F. Marshall","Oscar Mickelin","Amit Singer"],"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.","url_abs":"https://arxiv.org/abs/2406.05922v2","url_pdf":"https://arxiv.org/pdf/2406.05922v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"fast-expansion-into-harmonics-on-the-ball","repo_url":"https://github.com/oscarmickelin/fle_3d","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}