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Constructing measures with identical moments


Author: Alexey Kuznetsov
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 30E05; Secondary 30E20
DOI: https://doi.org/10.1090/proc/13585
Published electronically: May 4, 2017
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Abstract: When the moment problem is indeterminate, the Nevanlinna
parametrization establishes a bijection between the class of all measures having a prescribed set of moments and the class of Pick functions. The fact that all measures constructed through the Nevanlinna parametrization have identical moments follows from the theory of orthogonal polynomials and continued fractions. In this paper we explore the opposite direction: we construct a set of measures and we show that they all have identical moments, and then we establish a Nevanlinna-type parametrization for this set of measures. Our construction does not require the theory of orthogonal polynomials and it exposes the analytic structure behind the Nevanlinna parametrization.


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Additional Information

Alexey Kuznetsov
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
Email: kuznetsov@mathstat.yorku.ca

DOI: https://doi.org/10.1090/proc/13585
Keywords: Hamburger's indeterminate moment problem, Nevanlinna parametrization, entire functions, Cauchy Residue Theorem
Received by editor(s): August 2, 2016
Received by editor(s) in revised form: November 13, 2016
Published electronically: May 4, 2017
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society