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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
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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  • [1] D. Aharonov and H. S. Shapiro, Domains on which analytic functions satisfy quadrature identities, J. Analyse Math. 30 (1976), 39-73. MR 0447589 (56:5899)
  • [2] O. B. Bekken, B. K. Øksendal and A. Stray (Editors), Spaces of analytic functions, Lecture Notes in Math., vol. 512, Springer-Verlag, Berlin, 1976. MR 0463927 (57:3865)
  • [3] J. E. Brennan, Approximation in the mean by polynomials on non-Carathéodory domains, Ark. Mat. 15 (1977), 117-168. MR 0450566 (56:8859)
  • [4] P. J. Davis, The Schwartz function and its applications, Carus Math. Monographs, no. 17, Math. Assoc. Amer., 1974. MR 0407252 (53:11031)
  • [5] S. N. Mergelyan, On the completeness of systems of analytic functions, Amer. Math. Soc. Transl. (2) 19 (1962), 109-166. MR 24#A1410. MR 0131561 (24:A1410)
  • [6] M. Sakai, Analytic functions with finite Dirichlet integrals on Riemann surfaces, Acta Math. (to appear). MR 521461 (80d:30042)

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Keywords: Jordan domains, moment problems, polynomial approximation
Article copyright: © Copyright 1978 American Mathematical Society

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