Stability of factorization factors of the canonical factorization of Wiener–Hopf matrix functions

N. V. Adukova   ; V. L. Dilman

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 13 (2021) no. 1, pp. 5-13

Résumé

The problem of Wiener-Hopf factorization of matrix functions is one of the most demanded problems of mathematical analysis. However, its application is constrained by the fact that at the present time, there are no methods for constructively constructing factorization in the general case. In addition, the problem is far and by unstable, that is, a small perturbation of the original matrix function can lead to a change in the integer invariants of the problem (partial indices), and the factorization factors of the original and perturbed matrix functions may not be close. This means that the dependence of factors on perturbation is not continuous. The situation is complicated by the fact that factorization factors are found in a non-unique way, and therefore, before comparing factorizations, they need to be normalized. This problem is also not solved in the general case. In the well-known theorem of M.A. Shubin, the normalization problem is bypassed in the following way: it is proved that if the original and perturbed matrix functions have the same sets of partial indices, then their factorizations with close factorization factors exist. It is clear that in this case it is impossible to assess the degree of their proximity efficiently. In the proposed work, the theorem of M.A. Shubin is refined for the case when the original matrix function admits a canonical factorization. In this case, it is indicated how the canonical factorizations of two sufficiently close matrix functions should be normalized so that their factorization factors are also close enough. The main result of the work is to obtain explicit estimates, in terms of factorization of the original matrix function, for the absolute error in the approximate calculation of factors. The estimates are obtained by using the Toeplitz operator technique.

Détail

Keywords: Wiener–Hopf factorization, matrix functions, normalization of factorization, continuity of factors
Mots-clés : partial indices, error estimation.

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