{"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/a-nonuniform-fast-fourier-transform-based-on","title":"A nonuniform fast Fourier transform based on low rank approximation","arxiv_id":"1701.04492","date":"2017-01-17","proceeding":null,"authors":["Diego Ruiz-Antolin","Alex Townsend"],"abstract":"By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast, stable, and simple algorithm for computing the NUDFT that costs $\\mathcal{O}(N\\log N\\log(1/\\epsilon)/\\log\\!\\log(1/\\epsilon))$ operations based on the fast Fourier transform, where $N$ is the size of the transform and $0<\\epsilon <1$ is a working precision. Our key observation is that a NUDFT and DFT matrix divided entry-by-entry is often well-approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally-scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.","url_abs":"http://arxiv.org/abs/1701.04492v1","url_pdf":"http://arxiv.org/pdf/1701.04492v1.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":"a-nonuniform-fast-fourier-transform-based-on","repo_url":"https://github.com/MikaelSlevinsky/FastTransforms.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"a-nonuniform-fast-fourier-transform-based-on","repo_url":"https://github.com/JuliaApproximation/FastTransforms.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"a-nonuniform-fast-fourier-transform-based-on","repo_url":"https://github.com/UnofficialJuliaMirror/FastTransforms.jl-057dd010-8810-581a-b7be-e3fc3b93f78c","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"a-nonuniform-fast-fourier-transform-based-on","repo_url":"https://github.com/UnofficialJuliaMirrorSnapshots/FastTransforms.jl-057dd010-8810-581a-b7be-e3fc3b93f78c","is_official":0,"mentioned_in_paper":0,"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}