Inhomogeneous vector Riemann boundary value problem and convolutions equation on a finite interval

This paper proposes a method for investigating the inhomogeneous Riemann —Hilbert vector boundary value problem (also called the Riemann vector boundary value problem) in the Wiener algebra (to an convolution equation on a finite interval). The method consists in reducing the Riemann problem to a truncated Wiener —Hopf equation. It is shown that for the correct solvability of an inhomogeneous Riemann vector boundary value problem, it is necessary and sufficient to prove the uniqueness of the solution of the corresponding truncated homogeneous Wiener —Hopf equation. As a result of applying the method, new sufficient conditions for the existence of a canonical factorization of the function matrix in the Wiener algebra of order two are obtained.