{"paper":{"title":"Harmonic analysis of fractal measures induced by representations of a certain C$^*$-algebra","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Palle E. T. Jorgensen, Steen Pedersen","submitted_at":"1993-10-01T00:00:00Z","abstract_excerpt":"We describe a class of measurable subsets $\\Omega$ in $\\br^d$ such that $L^2(\\Omega)$ has an orthogonal basis of frequencies $e_\\lambda(x)=e^{i2\\pi\\lambda\\cdot x}(x\\in\\Omega)$ indexed by $\\lambda\\in\\Lambda\\subset\\br^d$. We show that such spectral pairs $(\\Omega ,\\Lambda)$ have a self-similarity which may be used to generate associated fractal measures $\\mu$ with Cantor set support. The Hilbert space $L^2(\\mu)$ does not have a total set of orthogonal frequencies, but a harmonic analysis of $\\mu$ may be built instead from a natural representation of the Cuntz C$^*$- algebra which is constructed "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9310233","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"}