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Inversion of the local Pompeiu transform. (English) Zbl 0723.44002

Let E be the collection of positive quotients of zeros of the Bessel function \(J_ 1\). We have proved elsewhere that if \(r_ 1,r_ 2>0\), \(r_ 1/r_ 2\not\in E\) and \(r_ 1+r_ 2<R\) then the map P: \(C^{\infty}(B(0,R))\to C^{\infty}(B(0,R-r_ 1))\oplus C^{\infty}(B(0,R-r_ 2))\), \(Pf=(g_ 1,g_ 2)\) given by \(g_ j(x)=\int_{B(0,r_ j)}f(x+y)dy\) is injective (where B(0,\(\rho\)) denotes the disk of center 0 and radius \(\rho\) in \({\mathbb{R}}^ 2)\). In this paper we find an explicit left inverse for the operator P. This inversion formula is a kind of adaptive convolution (in the sense that the formula remains the same in disks B(0,\(\rho\)), \(\rho <R\), but changes when we increase the radius \(\rho\) through a sequence \(\rho_ n\uparrow R)\), in contrast with the situation where \(R=\infty\) and \(r_ 1/r_ 2\) is badly approximated by elements of E, in which case we can find explicitly \(v_ 1\), \(v_ 2\in ({\mathbb{R}}^ 2)\) such that \(f=g_ 1*v_ 1+g_ 2*v_ 2\) (deconvolution).
We also show how to construct left inverses for other Pompeiu transforms, when averages over disks are replaced by averages over other sets, e.g. squares.

MSC:

44A12 Radon transform
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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