Description of Pompeiu sets in terms of approximations of their indicator
functions

O. A. Ochakovskaya

Processing math: 100%

	Description of Pompeiu sets in terms of approximations of their indicator functions
Author	O. A. Ochakovskaya ochakovskaja@yandex.ua Institute of Applied Mathematics and Mechanics NAS of Ukraine, Donetsk
Abstract	Let $H$ be an open upper half-space in $\mathbb R^n$ , $n\geq2$ , and assume that $A$ is a non-empty, open, bounded subset of $\mathbb R^n$ such that $\overline{A}\subset H$ and the exterior of $A$ is connected. Let $p\in[2, +\infty).$ It is proved that there is a nonzero function with zero integrals over all sets in $\mathbb R^n$ congruent to $A$ if and only if the indicator function of $A$ is the limit in $L^p(H)$ of a sequence of linear combinations of indicator functions of balls in $H$ with radii proportional to positive zeros of the Bessel function $J_{n/2}$ . The proportionality coefficient here is the same for all balls and depends only on $A$ .
Keywords	Pompeiu property; mean periodicity
Reference	1. Zalcman L. A bibliographic survey of the Pompeiu problem; in: B. Fuglede et al. (eds.)// Approximation by Solutions of Partial Differential Equations. – Dordrecht: Kluwer Academic Publishers, 1992. – P. 185– 194. 2. Zalcman L. Supplementary bibliography to ‘A bibliographic survey of the Pompeiu problem’; in Radon Transforms and Tomography// Contemp. Math. – 2001. – V.278. – P. 69–74. 3. Volchkov V.V. Integral Geometry and Convolution Equations. – Dordrecht: Kluwer Academic Publishers, 2003. – 454 p. 4. Volchkov V.V., Volchkov Vit.V. Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group. – London: Springer, 2009. – 671 p. 5. Volchkov V.V. New theorems on the mean for solutions of the Helmholtz equation// Russian Acad. Sci. Sb. Math. – 1994. – V.79. – P. 281–286. 6. Volchkov V.V. Theorems on ball means values in symmetric spaces// Sbornik: Math. – 2001. – V.192. – P. 1275–1296. 7. Ochakovskaya O.A. Precise characterizations of admissible rate of decrease of a non-trivial function with zero ball means// Sbornik: Math. – 2008. – V.199. – P. 45–65. 8. Korenev B.G. Introduction to the theory of Bessel functions. – Moscow: Nauka, 1971. – 288 p. 9. Levin B. Ya. Distribution of zeros of entire functions. – Providence, RI: Amer. Math. Soc., 1964. – 632 p. 10. Stein E., Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. – New Jersey: Princeton University Press, 1971. – 332 p.
Pages	142-149
Volume	39
Issue	2
Year	2013
Journal	Matematychni Studii
Full text of paper	PDF
Table of content of issue	HTML

Description of Pompeiu sets in terms of approximations of their indicator functions