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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization
of Carleman's uniqueness theorem
and a discrete Phragmén-Lindelöf theorem


Authors: B. Korenblum, A. Mascuilli and J. Panariello
Journal: Proc. Amer. Math. Soc. 126 (1998), 2025-2032
MSC (1991): Primary 30E05; Secondary 26E10
MathSciNet review: 1443835
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $d\mu \geq 0 $ be a Borel measure on $[0,\infty )$ and $A_{n}=\int \limits _{0}^{\infty }t^{n}d\mu (t) < \infty ~~(n=0,1,2,...)$ be its moments. T. Carleman found sharp conditions on the magnitude of $\{A_{n}\}_{0}^{\infty }$ for $d\mu $ to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if $A_{n}' =\int \limits _{0}^{\infty }t^{n} d\mu _{1} (t) $ are the moments of another measure, $d\mu _{1} \geq 0,$ with $\limsup \limits _{n\to \infty } |A_{n}-A_{n}'|^{\frac{1}{n}}=\rho <\infty ,$ then the measure $d\mu -d\mu _{1} $ is supported on the interval $[0,\rho ].$ This result generalizes both the Carleman theorem and a theorem of J. Mikusi\'{n}ski. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.


References [Enhancements On Off] (What's this?)

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Additional Information

B. Korenblum
Affiliation: Department of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222

A. Mascuilli
Affiliation: Department of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222

J. Panariello
Affiliation: Department of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04239-7
PII: S 0002-9939(98)04239-7
Keywords: Carleman's uniqueness theorem, quasianalyticity, Phragm\'{e}n-Lindel\"{o}f
Received by editor(s): June 13, 1996
Received by editor(s) in revised form: December 10, 1996
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1998 American Mathematical Society