Characterization of the Schur class in terms of the coefficients of a series on the Laguerre basis

Abstract

The classical Schur criterion answers the question of whether a function f given by its power series f(x)=∑k=0∞CkZk is a Schur function that is, holomorphic in a unit disk D and such that supz∈D | f (z) | ≤ 1. Regarding this criterion, there are a large number of completed results devoted to its generalizations and various applications, but, as it seems to us, there is no criterion for a complete description of the Schur class in terms of coefficients of orthogonal series on arbitrary complete orthonormal systems. In this paper, we formulate such criterion for a formal orthogonal series with complex coefficients based on the Laguerre system.