An Extension of the Geronimus Transformation for Orthogonal Matrix Polynomials on the Real Line
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- 作者：Juan Carlos García-Ardila ; Luis E. Garza…
- 关键词：Matrix orthogonal polynomials ; block Jacobi matrices ; matrix Geronimus transformation ; block Cholesky decomposition ; block LU decomposition ; quasi ; determinants
- 刊名：Mediterranean Journal of Mathematics
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：13
- 期：6
- 页码：5009-5032
- 全文大小：652 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
We consider matrix polynomials orthogonal with respect to a sesquilinear form \({\langle{\cdot,\cdot}\rangle_W}\), such that$$\langle{P(t)W(t),Q(t)W(t)}_{W}\rangle=\int_{\mathfrak{I}}P(t){\rm d}\muQ(t)^{T}, \quad P,Q \in \mathbb{P}^{p\times p}[t],$$where \({\mu}\) is a symmetric, positive definite matrix of measures supported in some infinite subset \({\mathfrak{I}}\) of the real line, and W(t) is a matrix polynomial of degree N. We deduce the integral representation of such sesquilinear forms in such a way that a Sobolev-type inner product appears. We obtain a connection formula between the sequences of matrix polynomials orthogonal with respect to \({\mu}\) and \({\langle{\cdot,\cdot}\rangle_W}\), as well as a relation between the corresponding block Jacobi and Hessenberg type matrices.KeywordsMatrix orthogonal polynomialsblock Jacobi matricesmatrix Geronimus transformationblock Cholesky decompositionblock LU decompositionquasi-determinantsThe work of Luis E. Garza was supported by Conacyt (México), Grant 156668. The work of Juan Carlos García-Ardila and Francisco Marcellán has been supported by Dirección General de Investigación, Científica y Técnica , Ministerio de Economía y Competitividad of Spain, Grant MTM2015-65888-C4-2-P.