{"paper":{"title":"A Paley-Wiener Type Theorem for Singular Measures on $\\mathbb{T}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CV","authors_text":"Eric S. Weber","submitted_at":"2017-09-21T21:44:22Z","abstract_excerpt":"For a fixed singular Borel probability measure $\\mu$ on $\\mathbb{T}$, we give several characterizations of when an entire function is the Fourier transform of some $f \\in L^2(\\mu)$. The first characterization is given in terms of criteria for sampling functions of the form $\\hat{f}$ when $f \\in L^2(\\mu)$. The second characterization is given in terms of criteria for interpolation of bounded sequences on $\\mathbb{N}_{0}$ by $\\hat{f}$. Both characterizations use the construction of Fourier series for $f \\in L^2(\\mu)$ demonstrated in Herr and Weber via the Kaczmarz algorithm and classical results"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.07522","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":""},"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"}