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Le problème de Pompeiu local. (Local Pompeiu problem). (French) Zbl 0668.30037
This paper extends our work from Isr. J. Math. 55, 267-288 (1986; Zbl 0624.31002). We show here that if a set \(\Omega\) has the Pompeiu property in \({\mathbb{R}}^ n\), generally it also satisfies the same property in balls of radius \(>R_ 0(\Omega)\). In order to avoid complicated definitions we give here two simple corollaries of the theorems proved. Hole Theorem. Let K be a convex closed subset of \({\mathbb{R}}^ n\), \(\Omega\) a cube and f a continuous function in \({\mathbb{R}}^ n\setminus K\). If for every rigid motion \(\sigma\) such that \(\sigma (\Omega)\cap K=\emptyset\) we have \(\int_{\sigma (\Omega)}fdx=0\), then \(f\equiv 0\) in \({\mathbb{R}}^ n\setminus K.\)
Morera Theorem. Let \(T_ 0\) be the boundary of a fixed triangle, \(T_ 0\subseteq \Delta(0,)=\) the open disk of radius \(1/2\) and center 0 in the complex plane. Let \(f\in C(\Delta (0,1))\) be such that \(\int_{T}f(z)dz=0\) for every T congruent (by a Euclidean motion) to \(T_ 0\), \(T\subseteq \Delta (0,1)\). Then f is holomorphic in the unit disk. A follow up paper with R. Gay and A. Yger will also appear in the same journal.
Reviewer: C.A.Berenstein

30E99 Miscellaneous topics of analysis in the complex plane
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35R99 Miscellaneous topics in partial differential equations
44A99 Integral transforms, operational calculus
Full Text: DOI
[1] C. A. Berenstein and R. Gay,A local version of the two-circles theorem, Isr. J. Math.55 (1986), 267–288. · Zbl 0624.31002 · doi:10.1007/BF02765026
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[4] A. M. Cormack and E. Quinto, A Radon transform through the origin in {ie165-1} and applications to the classical Darboux equation, Trans. Am. Math. Soc.260 (1980), 575–581. · Zbl 0444.44003
[5] S. Helgason,Groups and Geometric Analysis, Academic Press, New York, 1984. · Zbl 0543.58001
[6] L. Hörmander,The Analysis of Linear Partial Differential Operators, Vols. I and II, Springer-Verlag, Berlin, 1983.
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[8] B. Ja. Levin,Distribution of Zeros of Entire Functions, Am. Math. Soc., 1964. · Zbl 0152.06703
[9] L. Zalcman,Analyticity and the Pompeiu problem, Arch. Rat. Mech. Anal.47 (1972). · Zbl 0251.30047
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