Relative Asymptotics of Matrix Orthogonal Polynomials for Uvarov Perturbations: The Degenerate Case

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• 作者：Hossain O. Yakhlef ; Francisco Marcellán
• 关键词：Outer relative asymptotics ; matrix orthogonal polynomials ; perturbation of a matrix of measures
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：13
• 期：5
• 页码：3135-3153
• 全文大小：640 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1660-5454
• 卷排序：13

文摘

Let \({\alpha}\) be a square matrix of measures, and \({\left\{P_n(x; \alpha)\right\}_{n\geq 0}}\) the associated sequence of orthonormal matrix polynomials satisfying the three-term recurrence relation \({x P_n(x; \alpha) = A_{n+1}(\alpha)P_{n+1}(x; \alpha) + B_n(\alpha) P_n(x; \alpha) + A_n^{\ast}(\alpha)P_{n-1}(x; \alpha),}\)\({n \geq 0.}\) Let \({{\rm d}\beta(u) {\overset{ {\rm def} }{=}} {\rm d}\alpha(u) + M\delta(u - c)}\), where \({M}\) is a positive definite matrix, \({\delta (u - c)}\) is the Dirac measure supported at \({c}\) that is located outside the support of \({{\rm d}\alpha}\). We study the outer relative asymptotics of the sequence \({\left\{P_n(x; \beta)\right\}_{n\geq 0}}\) with respect to the sequence \({\left\{P_n(x; \alpha)\right\}_{n\geq 0}}\) under quite general assumption on the coefficients of the three-term recurrence relation \({\left\{A_{n}(\alpha) \right\}_{n\geq 0}}\) and \({\left\{B_{n}(\alpha) \right\}_{n\geq 0}}\).