{"paper":{"title":"A Fourier Frame for the Middle-Third Cantor Measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Carlos Cabrelli, Ursula Molter","submitted_at":"2018-12-13T21:59:40Z","abstract_excerpt":"In this paper we show that if $\\mu$ is any locally and uniformly $\\alpha$-dimensional measure supported on a $\\alpha$-quasi-regular set $E$, then $L^2(\\mu)$ admits a frame of exponentials. In particular, for the uniform middle third Cantor measure, $\\mu_C,$ our result shows that there exists a countable set $\\Lambda$ such that $\\{e^{2\\pi i t \\lambda}\\}_{\\lambda \\in \\Lambda}$ is a frame for $L^2(\\mu_C)$ (i.e. the measure $\\mu_C$ admits a generalized spectrum), answering an old outstanding question about the existence of a frame of exponentials for the space $L^2(\\mu_C)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.05708","kind":"arxiv","version":2},"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"}