Inversion of the local Pompeiu transform
type: Item
instance of: scholarly article
title: Inversion of the local Pompeiu transform
zbMATH Open document ID: 0723.44002
DOI: 10.1007/BF02796152
published in: Journal d'Analyse Mathématique
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 4192364
zbMATH Keywords: local Pompeiu transform
zbMATH Keywords: left inverse
zbMATH Keywords: inversion formula
zbMATH Keywords: adaptive convolution
zbMATH Keywords: deconvolution
author: Carlos A. Berenstein
author: R. Gay
author: A. Yger
reviewed by: Carlos A. Berenstein
MaRDI profile type: Publication
cites work: An inverse spectral theorem and its relation to the Pompeiu problem
cites work: A local version of the two-circles theorem
cites work: Le problème de Pompeiu local. (Local Pompeiu problem)
cites work: Q3351951
cites work: The ''three-squares'' theorem for continuous functions
cites work: Q3761007
cites work: Le problème de la deconvolution
cites work: Analytic Bezout identities
cites work: Pompeiu's problem on symmetric spaces
cites work: Fourier Transforms. (AM-19)
cites work: Spectral synthesis and the Pompeiu problem
cites work: A Radon Transform on Spheres Through the Origin in R n and Applications to the Darboux Equation
cites work: Q5811385
cites work: Q3671491
cites work: Q3331943
cites work: Spherical Harmonics
cites work: Q5660113
cites work: Q5562752
cites work: Analyticity and the Pompeiu problem
cites work: Offbeat Integral Geometry
full work available at URL: https://doi.org/10.1007/bf02796152
OpenAlex ID: W2093913251
description: scientific article; zbMATH DE number 4192364