{"title":"An Extremal Problem for the Bergman Kernel of Orthogonal Polynomials","authors":"S. Charpentier, N. Levenberg, F. Wielonsky","doi":"10.1007/s00365-023-09677-7","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(\\Gamma \\subset \\mathbb {C}\\)</span> be a curve of class <span>\\(C(1,\\alpha )\\)</span>. For <span>\\(z_{0}\\)</span> in the unbounded component of <span>\\(\\mathbb {C}\\setminus \\Gamma \\)</span>, and for <span>\\(n=1,2,...\\)</span>, let <span>\\(\\nu _n\\)</span> be a probability measure with <span>\\(\\mathop {\\textrm{supp}}\\nolimits (\\nu _{n})\\subset \\Gamma \\)</span> which minimizes the Bergman function <span>\\(B_{n}(\\nu ,z):=\\sum _{k=0}^{n}|q_{k}^{\\nu }(z)|^{2}\\)</span> at <span>\\(z_{0}\\)</span> among all probability measures <span>\\(\\nu \\)</span> on <span>\\(\\Gamma \\)</span> (here, <span>\\(\\{q_{0}^{\\nu },\\ldots ,q_{n}^{\\nu }\\}\\)</span> are an orthonormal basis in <span>\\(L^2(\\nu )\\)</span> for the holomorphic polynomials of degree at most <i>n</i>). We show that <span>\\(\\{\\nu _{n}\\}_n\\)</span> tends weak-* to <span>\\({{\\widehat{\\delta }}}_{z_{0}}\\)</span>, the balayage of the point mass at <span>\\(z_0\\)</span> onto <span>\\(\\Gamma \\)</span>, by relating this to an optimization problem for probability measures on the unit circle. Our proof makes use of estimates for Faber polynomials associated to <span>\\(\\Gamma \\)</span>.\n</p>","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":"150 1","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Constructive Approximation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00365-023-09677-7","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(\Gamma \subset \mathbb {C}\) be a curve of class \(C(1,\alpha )\). For \(z_{0}\) in the unbounded component of \(\mathbb {C}\setminus \Gamma \), and for \(n=1,2,...\), let \(\nu _n\) be a probability measure with \(\mathop {\textrm{supp}}\nolimits (\nu _{n})\subset \Gamma \) which minimizes the Bergman function \(B_{n}(\nu ,z):=\sum _{k=0}^{n}|q_{k}^{\nu }(z)|^{2}\) at \(z_{0}\) among all probability measures \(\nu \) on \(\Gamma \) (here, \(\{q_{0}^{\nu },\ldots ,q_{n}^{\nu }\}\) are an orthonormal basis in \(L^2(\nu )\) for the holomorphic polynomials of degree at most n). We show that \(\{\nu _{n}\}_n\) tends weak-* to \({{\widehat{\delta }}}_{z_{0}}\), the balayage of the point mass at \(z_0\) onto \(\Gamma \), by relating this to an optimization problem for probability measures on the unit circle. Our proof makes use of estimates for Faber polynomials associated to \(\Gamma \).
期刊介绍:
Constructive Approximation is an international mathematics journal dedicated to Approximations and Expansions and related research in computation, function theory, functional analysis, interpolation spaces and interpolation of operators, numerical analysis, space of functions, special functions, and applications.