{"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/sparse-averages-of-partial-sums-of-fourier","title":"Sparse Averages of Partial Sums of Fourier Series","arxiv_id":"1807.07636","date":"2018-07-19","proceeding":null,"authors":["Ethan Goolish","Robert S. Strichartz"],"abstract":"We study convergence properties of sparse averages of partial sums of Fourier series of continuous functions. By sparse averages, we are considering an increasing sequences of integers $n_0 < n_1 < n_2 < ...$ and looking at \\begin{equation*} \\tilde{\\sigma}_N(f)(t) = \\frac{1}{N+1}\\sum_{k=0}^{N}s_{n_k}(f)(t) \\end{equation*} to determine the necessary conditions on the sequence $\\{n_k\\}$ for uniform convergence. Among our results, we find that convergence is dependent on the sequence: we give a proof of convergence for the linear case, $n_k = pk$, for $p$ a positive integer, and present strong experimental evidence for convergence of the quadratic $n_k = k^2$ and cubic $n_k = k^3$ cases, but divergence for the exponential case, $n_k = 2^k$. We also present experimental evidence that if we replace the deterministic rules above by random processes with the same asymptotic behavior then almost surely the answer is the same.","url_abs":"http://arxiv.org/abs/1807.07636v2","url_pdf":"http://arxiv.org/pdf/1807.07636v2.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":"sparse-averages-of-partial-sums-of-fourier","repo_url":"https://github.com/egoolish/sparse","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}