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Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two

  Published:2012-12-04
 Printed: Jun 2014
  • Jianxun He,
    School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
  • Jinsen Xiao,
    Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China
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Abstract

Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.
Keywords: Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-Laplacian Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-Laplacian
MSC Classifications: 43A85, 44A12, 52A38 show english descriptions Analysis on homogeneous spaces
Radon transform [See also 92C55]
Length, area, volume [See also 26B15, 28A75, 49Q20]
43A85 - Analysis on homogeneous spaces
44A12 - Radon transform [See also 92C55]
52A38 - Length, area, volume [See also 26B15, 28A75, 49Q20]
 

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