An inverse problem associated with polynomials orthogonal on the unit circle
The paper reviews results for polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum. The goal is to formulate and solve an inverse problem for these polynomials. The authors use not only techniques of dynamical systems but some elementary methods from the theory of algebraic curves (e.g., generalized Jacobians). So it is possible to generalize some results obtained for polynomials with periodic recurrence coefficients to the quasi-periodic case. The quasi-periodicity of the recurrence coefficients is shown and measures are associated. Necessary and sufficient conditions relating quasi-periodicity and spectral measures of this type are given. The main problem that is posed and solved is a characterization of a special class of stationary ergodic sequences by working out explicit formulas for them. This method rests upon the Weyl \(m\)-functions with a special property that is also used to call special Jacobi operators \textit{reflectionless}. Finally, stability results are obtained by utilizing the Stahl-Totik theory of general orthogonal polynomials. Analogs for polynomials orthogonal on subsets of the real line are given.
scientific article; zbMATH DE number 1211618