A moment problem on Jordan domains

Sakai, Makoto

doi:10.1090/S0002-9939-1978-0470216-5

A moment problem on Jordan domains
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by Makoto Sakai PDF

Proc. Amer. Math. Soc. 70 (1978), 35-38 Request permission

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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