Laurent polynomial perturbations of linear functionals. An inverse problem
The authors continue their study of linear functionals \(L\) on the space of Laurent polynomials, generated by the sequence \(\{c_n\}_{n\in\mathbb{Z}}\) of moments \[ L(q)=\sum_{k=-n}^m c_kq_k, \qquad q(z)=\sum_{k=-n}^m q_kz^k, \] and corresponding Carathéodory functions \(F(z)=1+2\sum_{n\geq1} c_{-n}z^n\). In a previous paper they considered a perturbation \(L_R\) of \(L\) given by \[ L_R(q):=L\left(\frac{z+z^{-1}-\alpha-\bar\alpha}2\,q\right), \qquad \alpha\in\mathbb{C}. \] The main object under consideration in the present paper is the inverse transformation \(L_{R^{-1}}\), defined in a natural way. The authors obtain conditions for \(L_{R^{-1}}\) to be quasi-definite, as well as the relation between the corresponding Carathéodory functions. Several illuminating examples are analyzed.
scientific article; zbMATH DE number 5692978