Matrix Function Factorization And Its Applications In The Riemann Boundary Value Problems With The Reflection

This paper studies the factorization of a class of second order matrix function by combining the Wiener-Hopf factorization theory and analytical function boundary value problems. As applications, it discusses the Riemann boundary value problem with the reflection on the real axis and on the unit circle, and studies boundary value problems for bianalytic functions with the reflection on the unit circle.Firstly, this paper investigates the Wiener-Hopf factorization of a class of two order matrix function G(x). Precisely, the two order matrix function G(x) discussed should satisfy condition that the sum of the first column elements is equal to those of the second column elements, and the determinant of matrix function is 1. It discusses the relationship betiten the two order matrix function factorization and the corresponding Riemann boundary value problem, subsequently, it constructs the explicit Wiener-Hopf factorization expression of the two order matrix function by using the method of solving the Riemann boundary value problem.Secondly, as a first application, it considers the Riemann boundary value problem with the reflection on the real axis and on the unit circle. It establishes an equivalence relation of these boundary value problems and the vector-valued Riemann boundary value problem. With proper assumptions, it gains the explicit Wiener-Hopf factorization of coefficient matrix function, by virtue of the matrix function factorization method and the equivalent transformation of their solutions, it obtains the general solutions and the solvable conditions of the Riemann boundary value problem with the reflection on the real axis and on the unit circle.Finally, the Riemann boundary value problem for bianalytic functions with the reflection on the unit circle is proposed for the first time. By utilizing the factorization theorem on bianalytic functions, transformation method and growth order relation at infinite point between the bianalytic functions and its responding factorization, the original problem is equivalently transformed into the two vectorvalued Riemann boundary value problem, general solution and solution conditions of the original problem is given with the help of the matrix function factorization technique.