{"title":"逆问题、索博列夫-切比雪夫多项式和渐近性","authors":"","doi":"10.1007/s11253-024-02281-3","DOIUrl":null,"url":null,"abstract":"<p>Let (<em>u, v</em>) be a pair of quasidefinite and symmetric linear functionals with {<em>P</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> and {<em>Q</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {<em>R</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> as follows:</p> <p><span> <span>\\(\\begin{array}{cc}\\frac{{P}_{n+2}^{\\mathrm{^{\\prime}}}\\left(x\\right)}{n+2}+{b}_{n}\\frac{{P}_{n}^{\\mathrm{^{\\prime}}}\\left(x\\right)}{n}-{Q}_{n+1}\\left(x\\right)={d}_{n-1}\\left(x\\right),& n\\ge 1.\\end{array}\\)</span> </span></p> <p>We present necessary and sufficient conditions for {<em>R</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> to be orthogonal with respect to a quasidefinite linear functional <em>w.</em> In addition, we consider the case where {<em>P</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> and {<em>Q</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product</p> <p><span> <span>\\(\\langle p,q\\rangle s=\\underset{-1}{\\overset{1}{\\int }}pq{\\left(1-{x}^{2}\\right)}^{-1/2}dx+{\\uplambda }_{1}\\underset{-1}{\\overset{1}{\\int }}{p}^{\\mathrm{^{\\prime}}}{q}^{\\mathrm{^{\\prime}}}{\\left(1-{x}^{2}\\right)}^{1/2}dx+{\\uplambda }_{2}\\underset{-1}{\\overset{1}{\\int }}{p}^{\\mathrm{^{\\prime}}\\mathrm{^{\\prime}}}{q}^{\\mathrm{^{\\prime}}\\mathrm{^{\\prime}}}d\\mu \\left(x\\right),\\)</span> </span></p> <p>where <em>μ</em> is a positive Borel measure associated with <em>w</em> and λ<sub>1</sub><em>,</em> λ<sub>2</sub> <em>></em> 0; λ<sub>2</sub> is a linear polynomial of λ<sub>1</sub><em>.</em></p>","PeriodicalId":49406,"journal":{"name":"Ukrainian Mathematical Journal","volume":"18 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics\",\"authors\":\"\",\"doi\":\"10.1007/s11253-024-02281-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let (<em>u, v</em>) be a pair of quasidefinite and symmetric linear functionals with {<em>P</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> and {<em>Q</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {<em>R</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> as follows:</p> <p><span> <span>\\\\(\\\\begin{array}{cc}\\\\frac{{P}_{n+2}^{\\\\mathrm{^{\\\\prime}}}\\\\left(x\\\\right)}{n+2}+{b}_{n}\\\\frac{{P}_{n}^{\\\\mathrm{^{\\\\prime}}}\\\\left(x\\\\right)}{n}-{Q}_{n+1}\\\\left(x\\\\right)={d}_{n-1}\\\\left(x\\\\right),& n\\\\ge 1.\\\\end{array}\\\\)</span> </span></p> <p>We present necessary and sufficient conditions for {<em>R</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> to be orthogonal with respect to a quasidefinite linear functional <em>w.</em> In addition, we consider the case where {<em>P</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> and {<em>Q</em><sub><em>n</em></sub>}<sub><em>n≥</em>0</sub> are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product</p> <p><span> <span>\\\\(\\\\langle p,q\\\\rangle s=\\\\underset{-1}{\\\\overset{1}{\\\\int }}pq{\\\\left(1-{x}^{2}\\\\right)}^{-1/2}dx+{\\\\uplambda }_{1}\\\\underset{-1}{\\\\overset{1}{\\\\int }}{p}^{\\\\mathrm{^{\\\\prime}}}{q}^{\\\\mathrm{^{\\\\prime}}}{\\\\left(1-{x}^{2}\\\\right)}^{1/2}dx+{\\\\uplambda }_{2}\\\\underset{-1}{\\\\overset{1}{\\\\int }}{p}^{\\\\mathrm{^{\\\\prime}}\\\\mathrm{^{\\\\prime}}}{q}^{\\\\mathrm{^{\\\\prime}}\\\\mathrm{^{\\\\prime}}}d\\\\mu \\\\left(x\\\\right),\\\\)</span> </span></p> <p>where <em>μ</em> is a positive Borel measure associated with <em>w</em> and λ<sub>1</sub><em>,</em> λ<sub>2</sub> <em>></em> 0; λ<sub>2</sub> is a linear polynomial of λ<sub>1</sub><em>.</em></p>\",\"PeriodicalId\":49406,\"journal\":{\"name\":\"Ukrainian Mathematical Journal\",\"volume\":\"18 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-04-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ukrainian Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11253-024-02281-3\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ukrainian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11253-024-02281-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics
Let (u, v) be a pair of quasidefinite and symmetric linear functionals with {Pn}n≥0 and {Qn}n≥0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {Rn}n≥0 as follows:
We present necessary and sufficient conditions for {Rn}n≥0 to be orthogonal with respect to a quasidefinite linear functional w. In addition, we consider the case where {Pn}n≥0 and {Qn}n≥0 are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product
期刊介绍:
Ukrainian Mathematical Journal publishes articles and brief communications on various areas of pure and applied mathematics and contains sections devoted to scientific information, bibliography, and reviews of current problems. It features contributions from researchers from the Ukrainian Mathematics Institute, the major scientific centers of the Ukraine and other countries.
Ukrainian Mathematical Journal is a translation of the peer-reviewed journal Ukrains’kyi Matematychnyi Zhurnal, a publication of the Institute of Mathematics of the National Academy of Sciences of Ukraine.