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 be Jordan domains on the complex *z*-plane such that for every nonnegative integer *n*. Here *m* denotes two-dimensional Lebesgue measure. Does it follow that ? 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 and which imply that the above equality does not hold for some *n*.

**[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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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