{"paper":{"title":"Optimal Extrapolation Bounds for Sparse Fourier Sums","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.CA","math.NA"],"primary_cat":"cs.DS","authors_text":"Ruizhe Zhang","submitted_at":"2026-07-11T22:51:38Z","abstract_excerpt":"We prove an optimal extrapolation theorem for $k$-sparse Fourier sums over arbitrary real frequencies, without any separation assumption, bounding how large such a sum can be just outside an interval on which its energy is observed. For every $g(t)=\\sum_{j=1}^k v_j e^{i\\lambda_jt}$ with $\\lambda_j\\in\\mathbb R$ and every $x\\ge1$, $$\n  |g(x)|\\le k^{O(1)}\\exp(O(k\\mathop{\\mathrm{arcosh}} x))\\|g\\|_{L^2[-1,1]} . $$ In the endpoint regime, this refines to the explicit bound $$\n  |g(1+\\delta)|\\le O(k)\\exp(O(k\\sqrt\\delta))\\|g\\|_{L^2[-1,1]},\n  \\qquad 0\\le\\delta\\le1 . $$ This improves on the $\\exp(O(k^2\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10501","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.10501/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}