{"paper":{"title":"Fourier Series for Singular Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Eric S. Weber, John E. Herr","submitted_at":"2015-03-16T21:19:20Z","abstract_excerpt":"Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\\mu$ on $[0,1)$, every $f\\in L^2(\\mu)$ possesses a Fourier series of the form $f(x)=\\sum_{n=0}^{\\infty}c_ne^{2\\pi inx}$. We show that the coefficients $c_{n}$ can be computed in terms of the quantities $\\hat{f}(n) = \\int_{0}^{1} f(x) e^{-2\\pi i n x} d \\mu(x)$. We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $\\mu$-bandlimited."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.04856","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}