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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201501.png" /> be a Hausdorff topological space (cf. also [[Hausdorff space|Hausdorff space]]; [[Topological space|Topological space]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201502.png" /> a non-negative [[Radon measure|Radon measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201503.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201504.png" /> a [[Topological group|topological group]] of continuous self-mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201505.png" /> leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201506.png" /> invariant. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p1201509.png" /> denotes the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015011.png" />. A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015012.png" /> of compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015013.png" /> is said to have the Pompeiu property if the linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015014.png" /> given by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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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
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\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.
 
is injective.
  
A typical example occurs when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015017.png" /> is the [[Lebesgue measure|Lebesgue measure]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015018.png" /> is the Euclidean group of orientation-preserving rigid motions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015019.png" /> denote the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015021.png" /> its Fourier transform, which is an entire function of exponential type (cf. also [[Entire function|Entire function]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015022.png" />. In this case one can prove [[#References|[a7]]], [[#References|[a12]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015023.png" /> has the Pompeiu property if and only if
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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 valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015025.png" /> is the ball, (a2) can never be satisfied but if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015026.png" />, a pair of balls of radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015028.png" /> (the centre plays no role in this case), then it has the Pompeiu property if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015030.png" /> is the set of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015031.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015033.png" /> are positive roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015034.png" />. (Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015035.png" /> is the Bessel function of the first kind and order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015036.png" />, cf. [[Bessel functions|Bessel functions]].)
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015037.png" /> is an irreducible symmetric space of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015038.png" /> and non-compact type and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015039.png" /> is its group of orientation-preserving isometries [[#References|[a12]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015040.png" /> is a bounded open set with Lipschitz boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015042.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015043.png" /> is connected, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015044.png" />, then failure of the Pompeiu property for the singleton <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015045.png" /> is equivalent to the existence of an eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015046.png" /> for the overdetermined Neumann boundary value problem for the Euclidean Laplacian (cf. also [[Neumann boundary conditions|Neumann boundary conditions]])
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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]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015048.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015049.png" /> is a non-compact symmetric space of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015050.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015051.png" /> the invariant Laplacian in this case [[#References|[a4]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015052.png" /> for (a3) is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015053.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015054.png" /> 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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015055.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015056.png" />, condition (a2) is only known to be necessary, due to the failure of the [[Spectral synthesis|spectral synthesis]], [[#References|[a7]]]. For instance, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015057.png" />, except for elementary examples of the type of three squares with sides parallel to the axes and sizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015060.png" /> none of whose quotients is rational, one can show that if one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015061.png" /> to be, e.g., a rectangle, then (a2) is also sufficient for the Pompeiu property [[#References|[a6]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015062.png" />, find distributions of compact support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015063.png" /> such that
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015064.png" /></td> </tr></table>
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\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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015065.png" /> be the unit ball of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015066.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015067.png" /> be a pair of balls centred at the origin such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015069.png" />, then the values of all the integrals
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015070.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120150/p12015071.png" /> (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.
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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====
 
====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>
+
<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>

Latest revision as of 20:29, 10 January 2021

Let $X$ be a Hausdorff topological space (cf. also Hausdorff space; Topological space), $\mu$ a non-negative Radon measure on $X$, and $G$ a 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, 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 exponential type (cf. also Entire function) in $\mathbf{C}^n$. In this case one can prove [a7], [a12] that $\mathcal{K}$ has the Pompeiu property if and only if

(a2)

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.)

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 [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 $\alpha > 0$ for the overdetermined Neumann boundary value problem for the Euclidean Laplacian (cf. also 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 [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 [a4].

The natural conjecture that the existence of a solution $\alpha > 0$ for (a3) is equivalent to $\Omega$ being a Euclidean ball is usually called Schiffer's conjecture. For instance, [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 [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, [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 [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 [a3], [a5], [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 $r _ { 1 } + r _ { 2 } < 1$ 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 } < 1 ,\; j = 1,2, \end{equation*}

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

References

[a1] C. Berenstein, "On the converse to Pompeiu's problem" Noteas e Communicaçoes de Mat. Univ. Fed. Pernambuco (Brazil) , 73 (1976)
[a2] 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
[a3] C. Berenstein, E.V. Patrick, "Exact deconvolution for multiple operators" IEEE Proc. Multidimensional Signal Proc. , 78 (1990) pp. 723–734
[a4] C. Berenstein, M. Shahshahani, "Harmonic analysis and the Pompeiu problem" Amer. J. Math. , 105 (1983) pp. 1217–1229
[a5] C. Berenstein, D. Struppa, "Complex analysis and convolution equations" G.M. Henkin (ed.) , Encycl. Math. Sci. , 54 (1993) pp. 1–108
[a6] C. Berenstein, B.A. Taylor, "The three-squares theorem for continuous functions" Arch. Rat. Mech. Anal. , 63 (1977) pp. 253–259
[a7] L. Brown, B. Schreiber, B.A. Taylor, "Spectral synthesis and the Pompeiu problem" Ann. Inst. Fourier , 23 : 3 (1973) pp. 125–154
[a8] S. Casey, D. Walnut, "Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms" SIAM Review , 36 (1994) pp. 537–577
[a9] S. Williams, "A partial solution to the Pompeiu problem" Math. Ann. , 223 (1976) pp. 183–190
[a10] L. Zalcman, "Offbeat integral geometry" Amer. Math. Monthly , 87 (1980) pp. 161–175
[a11] 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)
[a12] C.A. Berenstein, L. Zalcman, "The Pompeiu problem in symmetric spaces" Comment. Math. Helvetici , 55 (1980) pp. 593–621
How to Cite This Entry:
Pompeiu problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pompeiu_problem&oldid=14506
This article was adapted from an original article by Carlos A. Berenstein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article