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Let $X$ be a Hausdorff topological space (cf. also [[Hausdorff space|Hausdorff space]]; [[Topological space|Topological space]]), $\mu$ a non-negative [[Radon measure|Radon measure]] on $X$, and $G$ a [[Topological group|topological group]] of continuous self-mappings of $X$ leaving $\mu$ invariant. For $x \in X$ and $g \in G$, $g.x$ denotes the action of $g$ on $X$. A family $\mathcal{K}$ of compact subsets of $X$ is said to have the Pompeiu property if the linear mapping $P : C ( X ) \rightarrow \Pi _ { K \in \mathcal{K} } C ( G )$ given by

\begin{equation} \tag{a1} P f ( g ) = \left( \int _ { g K } f d \mu \right) _ { K \in \mathcal{K} } , g \in G, \end{equation}

is injective.

A typical example occurs when $X = \mathbf{R} ^ { n }$, $\mu$ is the [[Lebesgue measure|Lebesgue measure]], and $G = M ( n )$ is the Euclidean group of orientation-preserving rigid motions. Let $\chi _ { K }$ denote the characteristic function of $K$ and $\widehat { \chi }_{K}$ its Fourier transform, which is an entire function of [[Function of exponential type|exponential type]] (cf. also [[Entire function|Entire function]]) in $\mathbf{C}^n$. In this case one can prove [[#References|[a7]]], [[#References|[a12]]] that $\mathcal{K}$ has the Pompeiu property if and only if

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015024.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a2)</td></tr></table>

When $\mathcal{K}$ is the ball, (a2) can never be satisfied but if $\mathcal{K} = \{ B _ { r _ { 1 } } , B _ { r _ { 2 } } \}$, a pair of balls of radii $r_1$ and $r_2$ (the centre plays no role in this case), then it has the Pompeiu property if and only if $r_{1} / r _ { 2 } \notin Z _ { n }$, where $Z_n$ is the set of fractions $\alpha / \beta$ for which $\alpha$ and $\beta$ are positive roots of $J _ { n  / 2} ( r ) = 0$. (Here, $J _ { n  / 2}$ is the Bessel function of the first kind and order $n / 2$, cf. [[Bessel functions|Bessel functions]].)

The key statement of the equivalence of the Pompeiu property with (a2) holds when $X$ is an irreducible symmetric space of rank $1$ and non-compact type and $G$ is its group of orientation-preserving isometries [[#References|[a12]]].

When $\Omega$ is a bounded open set with Lipschitz boundary $\partial \Omega$ in ${\bf R} ^ { n }$ such that $\mathbf{R} ^ { n } \backslash \overline { \Omega }$ is connected, and if $G = M ( n )$, then failure of the Pompeiu property for the singleton $\mathcal{K} = \{ \overline { \Omega } \}$ is equivalent to the existence of an eigenvalue $\alpha &gt; 0$ for the overdetermined Neumann boundary value problem for the Euclidean Laplacian (cf. also [[Neumann boundary conditions|Neumann boundary conditions]])

\begin{equation} \tag{a3} \left\{ \begin{array} { l } { \Delta u + \alpha u = 0 \quad \text { in } \Omega, } \\ { \frac { \partial u } { \partial n } = 0 \text { and } u = 1 \quad \text { on } \partial \Omega. } \end{array} \right. \end{equation}

It was shown by S. Williams [[#References|[a9]]] that if (a3) has a solution, then $\partial \Omega$ must be real-analytic, which allows for many positive examples of the Pompeiu property. The equivalence between (a3) and the failure of Pompeiu property and Williams' observation also holds when $X$ is a non-compact symmetric space of rank $1$ with $\Delta$ the invariant Laplacian in this case [[#References|[a4]]].

The natural conjecture that the existence of a solution $\alpha &gt; 0$ for (a3) is equivalent to $\Omega$ being a Euclidean ball is usually called Schiffer's conjecture. For instance, [[#References|[a1]]] contains the result that for convex planar sets the existence of infinitely many eigenvalues for (a3) implies that $\Omega$ is a disc. This inspired work of M. Agranovsky, C.A. Berenstein and P.C. Yang, N. Garofalo and F. Segala, T. Kobayashi, and others. See the excellent bibliographic survey [[#References|[a11]]] for the details on the progress made on this conjecture up to date (1998), as well as general background on the Pompeiu property.

For $X = G = {\bf R} ^ { n }$ with $n \geq 2$, condition (a2) is only known to be necessary, due to the failure of the [[Spectral synthesis|spectral synthesis]], [[#References|[a7]]]. For instance, for $n = 2$, except for elementary examples of the type of three squares with sides parallel to the axes and sizes $a$, $b$, $c$ none of whose quotients is rational, one can show that if one takes $K _ { 0 } \in \mathcal{K}$ to be, e.g., a rectangle, then (a2) is also sufficient for the Pompeiu property [[#References|[a6]]].

This case of the Pompeiu problem has many applications in image and signal processing and leads to the problem of deconvolution, that is, given a finite family $K _ { 1 } , \dots , K _ { \text{l} }$, find distributions of compact support $\nu _ { 1 } , \dots , \nu _ { \text{l} }$ such that

\begin{equation*} \nu _ { 1 } * \chi _ { K _ { 1 } } + \ldots + \nu _ { 1 }  { * } \chi _ { K _ { 1 } } = \delta, \end{equation*}

which amounts to finding a left inverse of the Pompeiu mapping (a1). See [[#References|[a3]]], [[#References|[a5]]], [[#References|[a8]]] for details.

There is a local variant of the Pompeiu problem. For instance, let $X = B ( 0,1 )$ be the unit ball of ${\bf R} ^ { n }$, let $B _ { r _ { 1 } } , B _ { r _ { 2 } }$ be a pair of balls centred at the origin such that $r _ { 1 } + r _ { 2 } &lt; 1$ and $r_1 / r _ { 2 } \notin \mathbf{Z} _ { n }$, then the values of all the integrals

\begin{equation*} \int _ { | x - a _ { j } | \leq r _ { j } } f ( x ) d x , \quad | a _ { j } | + r _ { j } &lt; 1 ,\; j = 1,2, \end{equation*}

are enough to determine any function $f \in C ( X )$ (the Berenstein–Gay theorem); see [[#References|[a2]]], [[#References|[a10]]], [[#References|[a11]]], for additional references, extensions, as well as relations to the deconvolution problem mentioned earlier.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  C. Berenstein,   "On the converse to Pompeiu's problem"  ''Noteas e Communicaçoes de Mat. Univ. Fed. Pernambuco (Brazil)'' , '''73'''  (1976)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  C. Berenstein,   "The Pompeiu problem, What's new"  R. Deville (ed.)  et al. (ed.) , ''Complex Analysis, Harmonic Analysis and Applications'' , ''Res. Notes Math.'' , '''347''' , Pitman  (1996)  pp. 1–11</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  C. Berenstein,   E.V. Patrick,   "Exact deconvolution for multiple operators"  ''IEEE Proc. Multidimensional Signal Proc.'' , '''78'''  (1990)  pp. 723–734</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Berenstein,   M. Shahshahani,   "Harmonic analysis and the Pompeiu problem"  ''Amer. J. Math.'' , '''105'''  (1983)  pp. 1217–1229</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  C. Berenstein,   D. Struppa,   "Complex analysis and convolution equations"  G.M. Henkin (ed.) , ''Encycl. Math. Sci.'' , '''54'''  (1993)  pp. 1–108</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  C. Berenstein,   B.A. Taylor,   "The three-squares theorem for continuous functions"  ''Arch. Rat. Mech. Anal.'' , '''63'''  (1977)  pp. 253–259</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  L. Brown,   B. Schreiber,   B.A. Taylor,   "Spectral synthesis and the Pompeiu problem"  ''Ann. Inst. Fourier'' , '''23''' :  3  (1973)  pp. 125–154</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  S. Casey,   D. Walnut,   "Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms"  ''SIAM Review'' , '''36'''  (1994)  pp. 537–577</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  S. Williams,   "A partial solution to the Pompeiu problem"  ''Math. Ann.'' , '''223'''  (1976)  pp. 183–190</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  L. Zalcman,   "Offbeat integral geometry"  ''Amer. Math. Monthly'' , '''87'''  (1980)  pp. 161–175</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  L. Zalcman,   "A bibliographic survey of the Pompeiu problem"  B. Fuglede (ed.)  et al. (ed.) , ''Approximation by solutions of partial differential equations'' , Kluwer Acad. Publ.  (1992)  pp. 185–194  (Addendum available from the author)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  C.A. Berenstein,   L. Zalcman,   "The Pompeiu problem in symmetric spaces"  ''Comment. Math. Helvetici'' , '''55'''  (1980)  pp. 593–621</td></tr></table>