{"paper":{"title":"Dirichlet Spaces Associated With Locally Finite Rooted Directed Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Deepak Kumar Pradhan, Sameer Chavan, Shailesh Trivedi","submitted_at":"2017-02-08T07:23:15Z","abstract_excerpt":"Let $\\mathscr T=(V, \\mathcal E)$ be a leafless, locally finite rooted directed tree. We associate with $\\mathscr T$ a one parameter family of Dirichlet spaces $\\mathscr H_q~(q \\geqslant 1)$, which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc $\\mathbb D$ in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel \\begin{eqnarray*} \\kappa_{\\mathscr H_q}(z, w) = \\sum_{n=0}^{\\infty}\\frac{(1)_n}{(q)_n}\\,{z^n \\overline{w}^n} ~P_{\\langle e_{\\mathsf{root}}\\rangle} + \\sum_{v \\in V_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02308","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"}