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Factorization of rational matrix functions and difference equations

100%
Rodríguez J. S., Campos L. F.
Concrete Operators
|
2013
|
tom 1
37-53
EN
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.
2
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A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

84%
Perälä A., Virtanen J. A., Wolf L.
Concrete Operators
|
2013
|
tom 1
28-36
EN
We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.
3
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On the Riemann-Hilbert problem in multiply connected domains

68%
Ryazanov V.
Open Mathematics
|
2016
|
tom 14
|
nr 1
13-18
EN
We proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthermore, it is shown that the dimension of the spaces of these solutions is infinite.
4
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Infinite dimension of solutions of the Dirichlet problem

61%
Ryazanov V.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.
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