
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 halfspace in $\mathbb R^n$, $n\geq2$, and assume that $A$ is
a nonempty, 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 
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Pages 
142149 
Volume 
39 
Issue 
2 
Year 
2013 
Journal 
Matematychni Studii 
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