
Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two
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 subLaplacian 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, subLaplacian Categories:43A85, 44A12, 52A38 