Some notes on the Radon transform and integral geometry

1992 ◽  
Vol 113 (1) ◽  
pp. 23-32 ◽  
Author(s):  
S. G. Gindikin
Keyword(s):  
Radon Transform ◽  
Integral Geometry

A Remark on Extensions of CR Functions from Hyperplanes

2008 ◽  
Vol 51 (1) ◽  
pp. 21-25
Author(s):  
Luca Baracco
Keyword(s):  
Radon Transform ◽  
Integral Geometry ◽  
Holomorphic Functions ◽  
Holomorphic Extension ◽  
General Statement ◽  
Cr Geometry ◽  
Cr Functions ◽  
Natural Context ◽  
Extension Of Functions

AbstractIn the characterization of the range of the Radon transform, one encounters the problem of the holomorphic extension of functions defined on ℝ2 \ Δℝ (where Δℝ is the diagonal in ℝ2) and which extend as “separately holomorphic” functions of their two arguments. In particular, these functions extend in fact to ℂ2 \ Δℂ where Δℂ is the complexification of Δℝ. We take this theorem from the integral geometry and put it in the more natural context of the CR geometry where it accepts an easier proof and amore general statement. In this new setting it becomes a variant of the celebrated “edge of the wedge” theorem of Ajrapetyan and Henkin.

The vertical slice transform on the unit sphere

2019 ◽  
Vol 22 (4) ◽  
pp. 899-917 ◽  
Author(s):  
Boris Rubin
Keyword(s):  
Unit Sphere ◽  
Radon Transform ◽  
Integral Geometry ◽  
Singular Value ◽  
Hypersingular Integrals ◽  
Inversion Formulas ◽  
Spherical Mean ◽  
Singular Value Decompositions ◽  
Vertical Slice ◽  
Class Of Functions

Abstract The vertical slice transform in spherical integral geometry takes a function on the unit sphere Sn to integrals of that function over spherical slices parallel to the last coordinate axis. This transform was investigated for n = 2 in connection with inverse problems of spherical tomography. The present article gives a survey of some methods which were originally developed for the Radon transform, hypersingular integrals, and the spherical mean Radon-like transforms, and can be adapted to obtain new inversion formulas and singular value decompositions for the vertical slice transform in the general case n ≥ 2 for a large class of functions.

scholarly journals On the Range of the Radon Transform onZnand the Related Volberg’s Uncertainty Principle

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Ahmed Abouelaz ◽  
Abdallah Ihsane ◽  
Takeshi Kawazoe
Keyword(s):  
Euclidean Space ◽  
Fourier Analysis ◽  
Uncertainty Principle ◽  
Exponential Type ◽  
Radon Transform ◽  
Integral Geometry ◽  
Discrete Radon Transform

We characterize the image of exponential type functions under the discrete Radon transformRon the latticeZnof the Euclidean spaceRn  n≥2. We also establish the generalization of Volberg's uncertainty principle onZn, which is proved by means of this characterization. The techniques of which we make use essentially in this paper are those of the Diophantine integral geometry as well as the Fourier analysis.

scholarly journals Integral geometry on discrete matrices

2021 ◽  
Vol 7 (3) ◽  
pp. 364-374
Author(s):  
Abdelbaki Attioui
Keyword(s):  
Radon Transform ◽  
Integral Geometry ◽  
Inversion Formula ◽  
Diophantine Equations ◽  
Support Theorem ◽  
Linear Diophantine Equations ◽  
Infinite Sets

Abstract In this note, we study the Radon transform and its dual on the discrete matrices by defining hyperplanes as being infinite sets of solutions of linear Diophantine equations. We then give an inversion formula and a support theorem.

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