{"paper":{"title":"Functional calculus on Venturi for Groups with Finite Propagation Speed","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Gordon Blower, Ian Doust","submitted_at":"2013-04-22T08:38:52Z","abstract_excerpt":"Let ${\\cal{M}}$ be a complete Riemannian manifold with Ricci curvature bounded below and Laplace operator $\\Delta$. The paper develops a functional calculus for the cosine family $\\cos(t\\sqrt {\\Delta})$ which is associated with waves that travel at unit speed. If $f$ is holomorphic on a Venturi shaped region, and $z^kf(z)$ is bounded for some positive integer $k$, then $f({\\sqrt \\Delta})$ defines a bounded linear operator on $L^p({\\cal{M}})$ for some $p>2$. For Jacobi hypergroups with invariant measure $m$ the generalized Fourier transform of $f\\in L^1(m)$ gives $\\hat f\\in H^\\infty (\\Sigma_\\om"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.5868","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"}