An Extension of the Geronimus Transformation for Orthogonal Matrix Polynomials on the Real Line

Juan Carlos García-Ardila; Luis E. Garza; Francisco Marcellán

doi:10.1007/s00009-016-0789-2

An Extension of the Geronimus Transformation for Orthogonal Matrix Polynomials on the Real Line

Juan Carlos García-Ardila, Luis E. Garza^*, Francisco Marcellán

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We consider matrix polynomials orthogonal with respect to a sesquilinear form ⟨ · , · ⟩ _W, such that ⟨P(t)W(t),Q(t)W(t)W⟩=∫_IP(t)dμQ(t)^T,P,Q∈P^p×p[t],where μ is a symmetric, positive definite matrix of measures supported in some infinite subset 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 μ and ⟨ · , · ⟩ _W, as well as a relation between the corresponding block Jacobi and Hessenberg type matrices.

Original language	English
Pages (from-to)	5009-5032
Number of pages	24
Journal	Mediterranean Journal of Mathematics
Volume	13
Issue number	6
DOIs	https://doi.org/10.1007/s00009-016-0789-2
Publication status	Published - 1 Dec 2016
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2016, Springer International Publishing.

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00009-016-0789-2

Cite this

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abstract = "We consider matrix polynomials orthogonal with respect to a sesquilinear form ⟨ · , · ⟩ W, such that ⟨P(t)W(t),Q(t)W(t)W⟩=∫IP(t)dμQ(t)T,P,Q∈Pp×p[t],where μ is a symmetric, positive definite matrix of measures supported in some infinite subset 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 μ and ⟨ · , · ⟩ W, as well as a relation between the corresponding block Jacobi and Hessenberg type matrices.",

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doi = "10.1007/s00009-016-0789-2",

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T1 - An Extension of the Geronimus Transformation for Orthogonal Matrix Polynomials on the Real Line

AU - García-Ardila, Juan Carlos

AU - Garza, Luis E.

AU - Marcellán, Francisco

PY - 2016/12/1

Y1 - 2016/12/1

N2 - We consider matrix polynomials orthogonal with respect to a sesquilinear form ⟨ · , · ⟩ W, such that ⟨P(t)W(t),Q(t)W(t)W⟩=∫IP(t)dμQ(t)T,P,Q∈Pp×p[t],where μ is a symmetric, positive definite matrix of measures supported in some infinite subset 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 μ and ⟨ · , · ⟩ W, as well as a relation between the corresponding block Jacobi and Hessenberg type matrices.

AB - We consider matrix polynomials orthogonal with respect to a sesquilinear form ⟨ · , · ⟩ W, such that ⟨P(t)W(t),Q(t)W(t)W⟩=∫IP(t)dμQ(t)T,P,Q∈Pp×p[t],where μ is a symmetric, positive definite matrix of measures supported in some infinite subset 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 μ and ⟨ · , · ⟩ W, as well as a relation between the corresponding block Jacobi and Hessenberg type matrices.

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An Extension of the Geronimus Transformation for Orthogonal Matrix Polynomials on the Real Line

Abstract

Bibliographical note

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