{"paper":{"title":"Szeg\\H{o}'s Condition on Compact subsets of $\\mathbb{C}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"G\\\"okalp Alpan","submitted_at":"2018-08-17T17:29:22Z","abstract_excerpt":"Let $K$ be a non-polar compact subset of $\\mathbb{C}$ and $\\mu_K$ be its equilibrium measure. Let $\\mu$ be a unit Borel measure supported on a compact set which contains the support of $\\mu_K$. We prove that a Szeg\\H{o} condition in terms of the Radon-Nikodym derivative of $\\mu$ with respect to $\\mu_K$ implies that $$\\inf_n \\frac{\\|P_n(\\cdot;\\mu)\\|_{L^2(\\mathbb{C};\\mu)}}{\\mathrm{Cap}(K)^n}>0.$$\n  We show that $\\frac{\\|P_n(\\cdot;\\mu_K)\\|_{L^2(\\mathbb{C};\\mu_K)}}{\\mathrm{Cap}(K)^n}\\geq 1$ for any compact non-polar set $K$. We also prove that under an additional assumption, unboundedness of the s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.05936","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"}