Expand all Collapse all | Results 1 - 3 of 3 |
1. CJM 2012 (vol 66 pp. 700)
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 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 Categories:43A85, 44A12, 52A38 |
2. CJM 2009 (vol 62 pp. 439)
On Hankel Forms of Higher Weights: The Case of Hardy Spaces In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by SundhÃ¤ll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.
Keywords:Hankel forms, Schattenâvon Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, MÃ¶bius group Categories:32A25, 32A35, 32A37, 47B35 |
3. CJM 2005 (vol 57 pp. 598)
Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group Differential operators $D_x$, $D_y$, and $D_z$ are formed using the
action of the $3$-dimensional discrete Heisenberg group $G$ on a set
$S$, and the operators will act on functions on $S$. The Laplacian
operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with
variable differences which can be associated to a unitary
representation of $G$ on the Hilbert space $L^2(S)$. Using techniques
from harmonic analysis and representation theory, we show that the
Laplacian operator is locally solvable.
Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operator Categories:43A85, 22D10, 39A70 |