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.
@article{bwmeta1.element.doi-10_2478_conop-2012-0005,
author = {J.S. Rodr\'\i guez and L.F. Campos},
title = {Factorization of rational matrix functions and difference equations},
journal = {Concrete Operators},
volume = {1},
year = {2013},
pages = {37-53},
zbl = {1320.47017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_conop-2012-0005}
}
J.S. Rodríguez; L.F. Campos. Factorization of rational matrix functions and difference equations. Concrete Operators, Tome 1 (2013) pp. 37-53. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_conop-2012-0005/
[1] H. Bart, I. Gohberg, M. A. Kaashoek, Minimal Factorization of Matrix and Operator Functions. Birkhäuser, 1979. | Zbl 0424.47001
[2] K. F. Clancey and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators. Birkhäuser, 1981.
[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] | Zbl 1049.47001
[4] S. E. Elaydi, An Introduction to Difference Equations. Springer, 1999. | Zbl 0930.39001
[5] G. S. Litvinchuk and I. M. Spitkovskii, Factorization of Measurable Matrix Functions. Operator Theory, Advances and Applications, Vol. 25, Birkhäuser, 1987.
[6] M. J. Morgado, Factorization of Rational Matrix Functions. University of Algarve, Portugal, 2006. (in portuguese)
[7] N. I. Muskhelishvili, Sigular Integral Equations. Dover Publications, 1992.