## 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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### References:

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