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.


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