Generalized factorization for Daniele-Khrapkov matrix functions -- explicit formulas
The problem of calculating the factors of a canonical generalized factorization of a class of non-rational matrix-valued functions of Daniele-Khrapkov type is solved. The method followed in the paper is a modification of that used for the investigation of conditions for the existence of canonical factorizations proposed by the authors in [ibid. 190, No. 1, 142-164 (1995; reviewed above)]. It is shown that the solutions of the Riemann-Hilbert problem with coefficient \(G\) which give the factors of \(G\) are related by a simple recursive system of algebraic equations to the homogeneous Riemann-Hilbert problem corresponding to the calculation of the functions in the kernel of a Toeplitz operator whose symbol \(\widetilde G\) possesses a non-canonical factorization. As a by-product of the analysis the authors solve explicitly an old problem stated by Daniele, that of determining a rational function \(R\) in the same group as \(G\), such that the factors in the factorization of \(RG\) belong to this group. These factors are then given by simple formulas analogous to those for the case where \(\rho\) is the square-root of the quotient of first degree polynomials. An example related to applications in diffraction theory is given.
non-rational matrix-valued functions of Daniele-Khrapkov type
scientific article; zbMATH DE number 765127