Method of Essential Functions and Its Application in the Theory of Wiener-Hopf factorization

Abstract

By analogy with the previously introduced notions of indices and essential polynomials of a finite sequence of matrices, we introduce indices and essential functions for an infinite sequence of matrix Fourier coefficients of an invertible matrix function, which belongs to the Wiener matrix algebra. Based on these notions, we propose a new proof of the existence of the Wiener — Hopf factorization of matrix functions by using only the methods of functional analysis. This method is applied to finding an explicit solution for the problem on the canonical factorization of a Laurent matrix polynomial of a special form. We obtain an effective sufficient criterion for the existence of a canonical factorization of strictly nonsingular second order matrix functions.