In the beginning of the XX century, Plemelj introduced the notion of factorization of matrix functions. If $A$ is a $n\times n$ regular matrix-function on the unit circle $\mathbb{T}$, by (left) factorization of $A$ relative to $\mathbb{T}$, we mean the following representation:
$$
A(t)=A_{+}(t)\Lambda(t)A_{-}(t), \quad t\in\mathbb{T},
$$
where
$$
\Lambda(t)=\mathrm{diag\,}(t^{\kappa_{1}},\dots,t^{\kappa_{n}}),
$$
with $\kappa_{1}\geq\dots\geq\kappa_{n}$, $\kappa_{i}\in\mathbb{Z}$ and $A_{+}^{\pm1}$ ($A_{-}^{\pm1}$) is analytic and regular in $\mathbb{T}_{+}$ ($\mathbb{T}_{-}$).
The matrix-function factorization find applications in many fields like diffraction theory, the theory of differential equations and the theory of singular integral operators.However, only for a few classes of matrices is known the explicit formulas for the factors of the factorization.
In our talk we will show a new method to obtain a factorization of rational matrix-functions. The constructed method is based on the relation between the general solution of an homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.
We also will provide some examples of factorization of rational matrix function, constructed with the developed factorization procedure.
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