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Estimates for the order of Nevanlinna matrices and a Berezanskii-type theorem

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Integral transforms, operational calculus Entire and meromorphic functions, and related topics Ordinary differential operators Special classes of linear operators

Published online by Cambridge University Press:  26 January 2019

Raphael Pruckner  and
Harald Woracek
Raphael Pruckner
Affiliation:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10/101 1040 Wien, AUSTRIA (raphael.pruckner@tuwien.ac.at; harald.woracek@tuwien.ac.at)
Harald Woracek
Affiliation:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10/101 1040 Wien, AUSTRIA (raphael.pruckner@tuwien.ac.at; harald.woracek@tuwien.ac.at)
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Abstract

We give an upper estimate for the order of the entire functions in the Nevanlinna parameterization of the solutions of an indeterminate Hamburger moment problem. Under a regularity condition this estimate becomes explicit and takes the form of a convergence exponent. Proofs are based on transformations of canonical systems and I.S.Kac' formula for the spectral asymptotics of a string. Combining with a lower estimate from previous work, we obtain a class of moment problems for which order can be computed. This generalizes a theorem of Yu.M.Berezanskii about spectral asymptotics of a Jacobi matrix (in the case that order is ⩽ 1/2).

Keywords

indeterminate moment problemorder of entire functionBerezanskii's theoremcanonical system

MSC classification

Primary: 44A60: Moment problems
Secondary: 30D15: Special classes of entire functions and growth estimates 37J99: None of the above, but in this section 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1637 - 1661
Copyright
Copyright © Royal Society of Edinburgh 2019 

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