Inhomogeneous Vector Riemann Boundary Value Problem and Convolutions Equation on a Finite Interval

In this paper, we develop a new method for studying the inhomogeneous vector Riemann-Hilbert boundary value problem (which is also called the Riemann boundary value problem) in the Wiener algebra of order two. The method consists in reducing the Riemann problem to a truncated Wiener-Hopf equation (to a convolution equation on a finite interval). The idea of the method was proposed by the author in a previous work. Here the method is applied to the inhomogeneous Riemann boundary value problem and to matrix functions of a more general form. The efficiency of the method is shown in the paper: new sufficient conditions for the existence of a canonical factorization of the matrix function in the Wiener algebra of order two are obtained. In addition, it was established that for the correct solvability of the inhomogeneous vector Riemann boundary value problem, it is necessary and sufficient to prove the uniqueness of the solution to the corresponding truncated homogeneous Wiener-Hopf equation.