Stability of Factorization Factors of Wiener–hopf Factorization of Matrix Function

Abstract

We consider the Wiener–Hopf factorization of two matrix functions A(t) and B(t) that are quite close in the norm of the Wiener algebra. The aim of this work is to study the question when the factorization factors of A(t), B(t) will be close enough to each other. This problem is of considerable interest in connection with the development of methods for approximate factorization of matrix functions. There are two main obstacles in the study of this problem: the instability of the partial indices of matrix functions and the non-uniqueness of their factorization factors. The problem was previously studied by M.A. Shubin, who showed that the stability of factorization factors takes place only in the case when A(t) and B(t) have the same partial indices. Then there is a factorization B(t) for which the factorization factors are sufficiently close to the factors of A(t). Theorem M.A. Shubin is non-constructive since it is not known when the partial indices of two close matrix functions will be the same, and the method for choosing the required Wiener–Hopf factorization of the matrix function B(t) is not indicated. To overcome these shortcomings, in the present paper we study the problem of normalization of the factorization in the stable case, describe all possible types of normalizations, and prove their stability under a small perturbation A(t). Now it is possible to find a constructive way of choosing the factorization of the perturbed matrix function, which guarantees the stability of the factorization factors.