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

1Savchuk, VV
Savchuk, MV
1Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2020, 11:9-15
Section: Mathematics
Language: Ukrainian
The classical Schur criterion answers the question of whether a function f given by its power series \[f(x)= \sum_{k=0}^{∞}C_{k}Z^{k}\] 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.
Keywords: Hardy space, holomorphic function, Laguerre basis, Schur class, Schur function

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