{"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-fourier-transforms-for-spherical-gauss","title":"Fast Fourier Transforms for Spherical Gauss-Laguerre Basis Functions","arxiv_id":"1604.05140","date":"2016-04-18","proceeding":null,"authors":["Jürgen Prestin","Christian Wülker"],"abstract":"Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)} (r^2) r^{l} Y_{lm}(\\vartheta,\\varphi)$, $|m| \\leq l < n \\in \\mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal basis of the space $L^{2}$ on $\\mathbb{R}^{3}$ with Gaussian weight $\\exp(-r^{2})$. These basis functions are used extensively, e.g., in biomolecular dynamic simulations. However, to the present, there is no reliable algorithm available to compute the Fourier coefficients of a function with respect to the SGL basis functions in a fast way. This paper presents such generalized FFTs. We start out from an SGL sampling theorem that permits an exact computation of the SGL Fourier expansion of bandlimited functions. By a separation-of-variables approach and the employment of a fast spherical Fourier transform, we then unveil a general class of fast SGL Fourier transforms. All of these algorithms have an asymptotic complexity of $\\mathcal{O}(B^{4})$, $B$ being the respective bandlimit, while the number of sample points on $\\mathbb{R}^{3}$ scales with $B^{3}$. This clearly improves the naive bound of $\\mathcal{O}(B^{7})$. At the same time, our approach results in fast inverse transforms with the same asymptotic complexity as the forward transforms. We demonstrate the practical suitability of our algorithms in a numerical experiment. Notably, this is one of the first performances of generalized FFTs on a non-compact domain. We conclude with a discussion, including the layout of a true $\\mathcal{O}(B^{3} \\log^{2} B)$ fast SGL Fourier transform and inverse, and an outlook on future developments.","url_abs":"http://arxiv.org/abs/1604.05140v3","url_pdf":"http://arxiv.org/pdf/1604.05140v3.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-fourier-transforms-for-spherical-gauss","repo_url":"https://github.com/cwuelker/SGLPack","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","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}