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Another step toward the solution of the Pompeiu problem in the plane. (English) Zbl 0818.35136
Let \(\Omega \subset \mathbb{R}^ 2\) be a bounded open set for which \(f \equiv 0\) is the only continuous function on \(\mathbb{R}^ 2\) such that (1) \(\int_{\sigma (\Omega)} f(x) dx = 0\) for every rigid motion \(\sigma\) of the plane. A bounded open set \(\Omega \subset \mathbb{R}^ 2\) is said to have the Pompeiu property if there exist no nontrivial continuous function on \(\mathbb{R}^ 2\) for which (1) holds. Several authors have determined bounded domains \(\Omega \subset \mathbb{R}^ 2\) having the Pompeiu property.
The main result of this paper is the following: Let \(\Omega \subset \mathbb{R}^ 2\) be a bounded simply-connected open set whose boundary is a closed simple curve parametrized by \(x(s) = (x_ 1(s), x_ 2(s))\), \(s \in [- \pi, \pi]\). Suppose that there exist \(M,N \in \mathbb{Z}\) and \(a_ k \in \mathbb{C}\), \(k = - M, \dots, N\), with \(a_ M\), \(a_ N \neq 0\), such that \[ x_ 1 (s) + x_ 2 (s) = \sum^ N_{k= -M} a_ k e^{iks}. \] Let \(x_ 1 (z)\), \(x_ 2 (z)\) be the analytic extension of \(x_ 1 (s)\) and \(x_ 2 (s)\) satisfying \((x_ 1'(z), x_ 2'(z))\neq (0,0) \in \mathbb{C}^ 2\) for every \(z \in \mathbb{C}\). Then \(\Omega\) has the Pompeiu property.
Reviewer: G.Anger (Berlin)

MSC:
35R30 Inverse problems for PDEs
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
35P05 General topics in linear spectral theory for PDEs
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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