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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

MSC:

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

Citations:

Zbl 0624.31002
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References:

[1] Berenstein, C. A.; Gay, R., A local version of the two-circles theorem, Isr. J. Math., 55, 267-288 (1986) · Zbl 0624.31002 · doi:10.1007/BF02765026
[2] Berenstein, C. A.; Zalcman, L., Pompeiu’s problem on symmetric spaces, Comment. Math. Helv., 55, 593-621 (1980) · Zbl 0452.43012 · doi:10.1007/BF02566709
[3] Brown, L.; Schreiber, B. M.; Taylor, B. A., Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier, 23, 125-154 (1973) · Zbl 0265.46044
[4] Cormack, A. M.; Quinto, E., A Radon transform through the origin in ℝ_\gp and applications to the classical Darboux equation, Trans. Am. Math. Soc., 260, 575-581 (1980) · Zbl 0444.44003 · doi:10.2307/1998023
[5] Helgason, S., Groups and Geometric Analysis (1984), New York: Academic Press, New York · Zbl 0543.58001
[6] Hörmander, L., The Analysis of Linear Partial Differential Operators (1983), Berlin: Springer-Verlag, Berlin · Zbl 0521.35002
[7] Lebedev, N. N., Special Functions and their Applications (1972), New York: Dover Publications, Inc., New York · Zbl 0271.33001
[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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