Inhomogeneous vector Riemann boundary value problem and convolutions equation on a finite interval (Q2027950)

A new method for studying the inhomogeneous vector Riemann-Hilbert boundary value problem in the Wiener algebra of order two is developed. The method consists in reducing the Riemann-Hilbert problem to a truncated Wiener-Hopf equation of the form \[ u(t)+\int\limits_{0}^{\tau}k(t-s)u(s)ds=f(t), \quad t\in(0,\tau). \] The idea of the method is described in the previous works of the author (see, e.g., [Sib. Èlektron. Mat. Izv. 15, 412--421 (2018; Zbl 1482.30103)]) and has been developed within the framework of this article. Here, the method is applied to the inhomogeneous Riemann boundary value problem and to matrix functions of a more general form. Some new sufficient conditions for the existence of a canonical factorization of a matrix function in a Wiener algebra of order two are obtained. In addition, it is 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.