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


Zbl 0624.31002
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.