# Factorization of rational matrix functions and difference equations

Concrete Operators (2013)

• Volume: 1, page 37-53
• ISSN: 2299-3282

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## Abstract

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In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.

## How to cite

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J.S. Rodríguez, and L.F. Campos. "Factorization of rational matrix functions and difference equations." Concrete Operators 1 (2013): 37-53. <http://eudml.org/doc/266700>.

@article{J2013,
abstract = {In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.},
author = {J.S. Rodríguez, L.F. Campos},
journal = {Concrete Operators},
keywords = {explicit factorization; rational matrix functions; factorization of matrix functions; Riemann-Hilbert boundary value problem; linear difference equations},
language = {eng},
pages = {37-53},
title = {Factorization of rational matrix functions and difference equations},
url = {http://eudml.org/doc/266700},
volume = {1},
year = {2013},
}

TY - JOUR
AU - J.S. Rodríguez
AU - L.F. Campos
TI - Factorization of rational matrix functions and difference equations
JO - Concrete Operators
PY - 2013
VL - 1
SP - 37
EP - 53
AB - In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.
LA - eng
KW - explicit factorization; rational matrix functions; factorization of matrix functions; Riemann-Hilbert boundary value problem; linear difference equations
UR - http://eudml.org/doc/266700
ER -

## References

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1. [1] H. Bart, I. Gohberg, M. A. Kaashoek, Minimal Factorization of Matrix and Operator Functions. Birkhäuser, 1979. Zbl0424.47001
2. [2] K. F. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators. Birkhäuser, 1981.
3. [3] I. Gohberg, M. A. Kaashoek and I. M. Spitkovsky, An Overview of Matrix Factorization. Operator Theory, Advances and Applications, Vol. 141, 1-102,Birkhäuser, 2003 [WoS] Zbl1049.47001
4. [4] S. E. Elaydi, An Introduction to Difference Equations. Springer, 1999. Zbl0930.39001
5. [5] G. S. Litvinchuk and I. M. Spitkovskii, Factorization of Measurable Matrix Functions. Operator Theory, Advances and Applications, Vol. 25, Birkhäuser, 1987.
6. [6] M. J. Morgado, Factorization of Rational Matrix Functions. University of Algarve, Portugal, 2006. (in portuguese)
7. [7] N. I. Muskhelishvili, Sigular Integral Equations. Dover Publications, 1992.

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