{"paper":{"title":"Moments of the weighted Cantor measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Alexander W. N. Riasanovsky, Steven N. Harding","submitted_at":"2019-08-14T21:44:56Z","abstract_excerpt":"Based on the seminal work of Hutchinson, we investigate properties of {\\em $\\alpha$-weighted Cantor measures} whose support is a fractal contained in the unit interval. Here, $\\alpha$ is a vector of nonnegative weights summing to $1$, and the corresponding weighted Cantor measure $\\mu^\\alpha$ is the unique Borel probability measure on $[0,1]$ satisfying $ \\mu^\\alpha(E) = \\sum_{ n=0 }^{N-1} \\alpha_n\\mu^\\alpha( \\varphi_n^{-1}(E) )$ where $\\varphi_n: x\\mapsto (x+n)/N$. In Sections 1 and 2 we examine several general properties of the measure $\\mu^\\alpha$ and the associated Legendre polynomials in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05358","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/1908.05358/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"}