AN EQUATION RELATED TO THE RIEMANN PROBLEM WITH A RATIONAL COEFFICIENT THAT IS INVERTIBLE IN ONE OF THE SUBRINGS OF A FACTORIZATION PAIR

Abstract

We report the results of a continuation of the study on the solvability of an abstract equation involving two rational functions from subrings. By the equation can be describe the boundary conditions of a problem related to the Riemann (Riemann-Hilbert, Riemann-Hilbert-Privalov) boundary value problem in the theory of analytic functions. We consider the case when the coefficient is a rational function belonging to one of the subrings of a factorization pair of the ring and is invertible in this subring. The equation is solved in the entire complex plane, and the boundary condition is obtained by restricting it to the closed real axis. We employ an approach that is distinguished by its algebraicity and simplifies the proof. This approach combines fundamental principles of ring theory and functional analysis, the theory of linear operators, and equations in rings with factorization pairs. A theorem on existence and uniqueness is established, along with general formulas for the desired solutions. An illustrative example is provided. The method used avoids the theory of Cauchy and Fourier-type integrals, the requirement of function Holder’s and the concept of index.