Connections between the Carathéodory-Toeplitz and the Nehari extension problems: The discrete scalar case
Let \({\mathcal W}\) denote the Wiener algebra of functions defined on the unit circle with Fourier coefficients in \(\ell_1 (\mathbb{Z})\) (i.e., \(f(z) = \sum_{n \in \mathbb{Z}} f_nz^n\), with \(\sum_{n \in \mathbb{Z}} |f_n |< \infty\)), and let us denote by \({\mathcal W}_{+}\) the subalgebra of functions with \(f_n = 0\) for all \(n < 0\) and by \({\mathcal W}_{-}\) the subalgebra of functions with \(f_{n}=0\) for all \(n > 0\). A function \(f \in {\mathcal W}\) is said to have a Weiner-Hopf factorization if it can be written as \(f = f_{+} f_{-}\), where both \(f_{+}\) and \(1/ f_{+}\) belong to \({\mathcal W}_{+}\) and both \(f_{-}\) and \(1/ f_{-}\) belong to \({\mathcal W}_{-}\). The Carathéodory-Toeplitz and the Nehary extension problems (resp.) both consists in the description of the class of functions \(f\in {\mathcal W}\) whose Fourier coefficients satisfy certain restrictions (they interpolate a fixed special sequence of values) and their solution depends on certain sequences of matrix functions \(T_n(z)\) and \(H_n(z)\) (resp.). In the paper a relation is stated which exists between the \(T_n\) and the \(H_n\), which is given by two complicated formulas (see Theorem 3.1 of the paper). The problem is firstly solved for the rational functions which satisfy the corresponding extension problems and then the authors use an approximation theoretical argument to prove the main result in the general case. The paper is continuation of a previous one by the same authors [see: Integral Equations Oper. Theory 26, No. 3, 249-272 (1996; Zbl 0871.47015)].
scientific article; zbMATH DE number 1507904