The weighted Nehari-Dym-Gohberg problem (Q1402349)

The matrix version of Nehari's problem adapted to the Wiener class setting is considered in this paper in the situation when two weights are included. Solvability conditions and parametric descriptions of the solutions are given. Given a pair \(p\), \(q\) of natural numbers, let \(\{c_n\}_{n=1}^\infty\) be a sequence of \(p\times q\) matrices with complex entries such that \(\sum_{n=1}^\infty| c_n| <\infty\), and let \(w_1\) and \(w_2\) be two weights on the unit circle of the complex plane whose values are either complex numbers or complex matrices with \(q\times q\) and \(p\times p\) entries, respectively. The authors deal with the problem of determining whether there exists a matrix-valued function \(\Psi\) on the unit circle whose negative Fourier coefficients are the prescribed matrices \(c_n\) and such that \(\Psi\) is a Wiener function and the supremum norm of \(w_2^{-1/2} \Psi w_1^{-1/2}\) is less than 1. In this note the scalar weight case of the weighted Nehari-Dym-Gohberg problem is related in four situations according to whether the weights \(w_1\), \(w_2\) are of Szegö type or not. In each of those situations the authors establish conditions for the existence of solutions and present a complete description of all solutions. The nonweighted problem is also discussed as a preliminary and particular case. The results are achieved via the Cotlar-Sadosky algebraic scattering system methods and by the use of the Arov-Grossman functional model.