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 be a Borel measure on and be its moments. T. Carleman found sharp conditions on the magnitude of for to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if are the moments of another measure, with then the measure is supported on the interval 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.

**[C]**T. Carleman,*Sur le Probléme des Moments*, Comptus Rendus Acad. Sci. Paris**174**(1922), 1680.**[K]**B. I. Korenbljum,*Quasianalytic classes of functions in a circle*, Dokl. Akad. Nauk SSSR**164**(1965), 36–39 (Russian). MR**0212199****[Ma]**S. Mandelbrojt,*Séries adhérentes, régularisation des suites, applications*, Gauthier-Villars, Paris, 1952 (French). MR**0051893****[Mar]**A. I. Markushevich,*Theory of functions of a complex variable. Vol. I, II, III*, Second English edition, Chelsea Publishing Co., New York, 1977. Translated and edited by Richard A. Silverman. MR**0444912****[Mi]**Jan G.-. Mikusiński,*Remarks on the moment problem and a theorem of Picone*, Colloquium Math.**2**(1951), 138–141. MR**0043150****[PW]**R. Paley and N. Wiener,*Fourier Transforms in the Complex Domain*, American Mathematical Society Colloquium Publications, vol. XIX, Providence, R.I., 1934. CMP**97:13**

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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

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