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

Author O. A. Ochakovskaya
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
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Pages 142-149
Volume 39
Issue 2
Year 2013
Journal Matematychni Studii
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