{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:RDJK73WYKQ6SSGN355CKUHMA4V","short_pith_number":"pith:RDJK73WY","schema_version":"1.0","canonical_sha256":"88d2afeed8543d2919bbef44aa1d80e5619638a2b9a2aa36010867536cdacabb","source":{"kind":"arxiv","id":"1602.02745","version":2},"attestation_state":"computed","paper":{"title":"On measures which generate the scalar product in a space of rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.FA"],"primary_cat":"math.CV","authors_text":"Victor Katsnelson","submitted_at":"2016-02-07T14:49:57Z","abstract_excerpt":"Let $z_1,z_2,\\,\\ldots\\,,z_n$ be pairwise different points of the unit disc and $\\mathscr{L}(z_1,z_2,\\,\\ldots\\,z_n)$ be the linear space generated by the rational fractions $\\frac{1}{t-z_1} , \\frac{1}{t-z_2} , \\cdots\\ , \\frac{1}{t-z_n}\\cdot$ Every non-negative measure $\\sigma$ on the unit circle $\\mathbb{T}$ generates the scalar product \\[\\langle\\,f\\,,\\,g\\,\\rangle_{\\!_{L^2_\\sigma}} =\\int\\limits_{\\mathbb{T}}f(t)\\,\\bar{g(t)}\\,\\sigma(dt), \\quad \\forall\\,f,g\\,\\in\\,L^2_\\sigma.\\] The measures $\\sigma$ are described which satisfy the condition \\[\\langle\\,f\\,,\\,g\\,\\rangle_{\\!_{L^2_\\sigma}}= 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measures which generate the scalar product in a space of rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.FA"],"primary_cat":"math.CV","authors_text":"Victor Katsnelson","submitted_at":"2016-02-07T14:49:57Z","abstract_excerpt":"Let $z_1,z_2,\\,\\ldots\\,,z_n$ be pairwise different points of the unit disc and $\\mathscr{L}(z_1,z_2,\\,\\ldots\\,z_n)$ be the linear space generated by the rational fractions $\\frac{1}{t-z_1} , \\frac{1}{t-z_2} , \\cdots\\ , \\frac{1}{t-z_n}\\cdot$ Every non-negative measure $\\sigma$ on the unit circle $\\mathbb{T}$ generates the scalar product \\[\\langle\\,f\\,,\\,g\\,\\rangle_{\\!_{L^2_\\sigma}} =\\int\\limits_{\\mathbb{T}}f(t)\\,\\bar{g(t)}\\,\\sigma(dt), \\quad \\forall\\,f,g\\,\\in\\,L^2_\\sigma.\\] The measures $\\sigma$ are described which satisfy the condition \\[\\langle\\,f\\,,\\,g\\,\\rangle_{\\!_{L^2_\\sigma}}= 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