{"title":"Density of Simple Partial Fractions with Poles on a Circle in Weighted Spaces for a Disk and a Segment","authors":"M. A. Komarov","doi":"10.1134/s1063454124010072","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper, we study approximation properties of simple partial fractions (logarithmic derivatives of algebraic polynomials), all of whose poles lie on the unit circle. There are obtained criteria for the density of these fractions in classical integral spaces: in the spaces of functions summable with degree <i>p</i> on the unit segment with ultraspherical weight and (weighted) Bergman spaces, analytic in the unit disk and summable with degree <i>p</i> over the disk area. The well-known criteria of Chui and Newman and Abakumov, Borichev, and Fedorovsky for Bergman spaces with <i>p</i> = 1 and <i>p</i> = 2, respectively, are generalized by the obtained results to the case of an arbitrary exponent <i>p</i> > 0.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"28 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik St Petersburg University-Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1063454124010072","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study approximation properties of simple partial fractions (logarithmic derivatives of algebraic polynomials), all of whose poles lie on the unit circle. There are obtained criteria for the density of these fractions in classical integral spaces: in the spaces of functions summable with degree p on the unit segment with ultraspherical weight and (weighted) Bergman spaces, analytic in the unit disk and summable with degree p over the disk area. The well-known criteria of Chui and Newman and Abakumov, Borichev, and Fedorovsky for Bergman spaces with p = 1 and p = 2, respectively, are generalized by the obtained results to the case of an arbitrary exponent p > 0.
期刊介绍：
Vestnik St. Petersburg University, Mathematics is a journal that publishes original contributions in all areas of fundamental and applied mathematics. It is the prime outlet for the findings of scientists from the Faculty of Mathematics and Mechanics of St. Petersburg State University. Articles of the journal cover the major areas of fundamental and applied mathematics. The following are the main subject headings: Mathematical Analysis; Higher Algebra and Numbers Theory; Higher Geometry; Differential Equations; Mathematical Physics; Computational Mathematics and Numerical Analysis; Statistical Simulation; Theoretical Cybernetics; Game Theory; Operations Research; Theory of Probability and Mathematical Statistics, and Mathematical Problems of Mechanics and Astronomy.
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