On explicit Wiener–Hopf factorization of 2 × 2 matrices in a vicinity of a given matrix

As it is known, the existence of the Wiener–Hopf factorization for a given matrix is a well-studied problem. Severe difficulties arise, however, when one needs to compute the factors approximately and obtain the partial indices. This problem is very important in various engineering applications and, therefore, remains to be subject of intensive investigations. In the present paper, we approximate a given matrix function and then explicitly factorize the approximation regardless of whether it has stable partial indices. For this reason, a technique developed in the Janashia–Lagvilava matrix spectral factorization method is applied. Numerical simulations illustrate our ideas in simple situations that demonstrate the potential of the method.

Rank-deficient spectral factorization and wavelets completion problem

A simple constructive proof of polynomial matrix spectral factorization theorem is presented in the rank-deficient case. It is then used to provide an elementary solution to the wavelets completion problem.

An elementary proof of the polynomial matrix spectral factorization theorem

A very simple proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line), which relies on elementary complex analysis and linear algebra, is presented.