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A moment problem on Jordan domains


Author: Makoto Sakai
Journal: Proc. Amer. Math. Soc. 70 (1978), 35-38
MSC: Primary 30A80
DOI: https://doi.org/10.1090/S0002-9939-1978-0470216-5
MathSciNet review: 0470216
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Abstract: Let $ {D_1},{D_2}$ be Jordan domains on the complex z-plane such that $ {\smallint _{{D_1}}}{z^n}dm = {\smallint _{{D_2}}}{z^n}dm$ for every nonnegative integer n. Here m denotes two-dimensional Lebesgue measure. Does it follow that $ {D_1} = {D_2}$? This moment problem on Jordan domains was posed by H. S. Shapiro [2, p. 193, Problem 1]. In this paper we construct a counterexample and study conditions on $ {D_1}$ and $ {D_2}$ which imply that the above equality does not hold for some n.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0470216-5
Keywords: Jordan domains, moment problems, polynomial approximation
Article copyright: © Copyright 1978 American Mathematical Society

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