scholarly journals The Radon Transforms on the Generalized Heisenberg Group

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Tianwu Liu ◽  
Jianxun He
Keyword(s):  
Fourier Transform ◽  
Differential Operator ◽  
Heisenberg Group ◽  
Radon Transform ◽  
Inversion Formula ◽  
The Other ◽  
Radon Transforms

Let Hna be the generalized Heisenberg group. In this paper, we study the inversion of the Radon transforms on Hna. Several kinds of inversion Radon transform formulas are established. One is obtained from the Euclidean Fourier transform; the other is derived from the differential operator with respect to the center variable t. Also by using sub-Laplacian and generalized sub-Laplacian we deduce an inversion formula of the Radon transform on Hna.

An Inversion Formula of the Radon Transform on the Heisenberg Group

2004 ◽  
Vol 47 (3) ◽  
pp. 389-397 ◽  
Author(s):  
Jianxun He
Keyword(s):  
Heisenberg Group ◽  
Radon Transform ◽  
Inversion Formula ◽  
Weak Sense

AbstractIn this paper we give an inversion formula of the Radon transform on the Heisenberg group by using the wavelets defined in [3]. In addition, we characterize a space such that the inversion formula of the Radon transform holds in the weak sense.

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

2014 ◽  
Vol 66 (3) ◽  
pp. 700-720 ◽  
Author(s):  
Jianxun He ◽  
Jinsen Xiao
Keyword(s):  
Fourier Transform ◽  
Lie Group ◽  
Radon Transform ◽  
Wavelet Transforms ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
Nilpotent Lie Group ◽  
Differentiability Property ◽  
3 Dimensional ◽  
Group Fourier Transform

AbstractLet F2n;2 be the free nilpotent Lie group of step two on 2n generators, and let P denote the affine automorphism group of F2n;2. In this article the theory of continuous wavelet transformon F2n;2 associated with P is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on F2n;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 the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if n = 1, F2;2 is the 3-dimensional Heisenberg group H1, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on F2;2. This result cannot be extended to the case n ≥ 2.

INVERSION OF THE RADON TRANSFORM ASSOCIATED WITH THE CLASSICAL DOMAIN OF TYPE ONE

2005 ◽  
Vol 16 (08) ◽  
pp. 875-887 ◽  
Author(s):  
JIANXUN HE ◽  
HEPING LIU
Keyword(s):  
Wavelet Transform ◽  
Differential Operator ◽  
Lie Group ◽  
Radon Transform ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Weak Sense ◽  
Continuous Wavelet ◽  
Nilpotent Lie Group ◽  
Classical Domain

Let D(Ω,Φ) be the unbounded realization of the classical domain [Formula: see text] of type one. In general, its Šilov boundary [Formula: see text] is a nilpotent Lie group of step two. In this article we define the Radon transform on [Formula: see text], and obtain an inversion formula [Formula: see text] in terms of a determinantal differential operator. Moreover, we characterize a subspace of [Formula: see text] on which the Radon transform is a bijection. By use of the suitable continuous wavelet transform we establish a new inversion formula of the Radon transform in weak sense without the assumption of differentiability.

scholarly journals Littlewood-Paley g-function and Radon Transform on the Heisenberg Group

2018 ◽  
Vol 11 (1) ◽  
pp. 138
Author(s):  
Zheng Fang ◽  
Jianxun He
Keyword(s):  
Heisenberg Group ◽  
Radon Transform ◽  
Unitary Operator ◽  
Radon Transforms ◽  
Inversion Formulas ◽  
Poisson Integrals

In this paper, we consider Radon transform on the Heisenberg group $\textbf{H}^{n}$, and obtain new inversion formulas via dual Radon transforms and Poisson integrals. We prove that the Radon transform is a unitary operator from Sobelov space $W$ into $L^{2}(\textbf{H}^{n})$. Moreover, we use the Radon transform to define the Littlewood-Paley $g$-function on a hyperplane and obtain the Littlewood-Paley theory.

scholarly journals On an analytic description of the α-cosine transform on real Grassmannians

2016 ◽  
Vol 18 (02) ◽  
pp. 1550025 ◽  
Author(s):  
Semyon Alesker ◽  
Dmitry Gourevitch ◽  
Siddhartha Sahi
Keyword(s):  
Differential Operator ◽  
Radon Transform ◽  
Open Problem ◽  
Analytic Description ◽  
Radon Transforms ◽  
Invariant Differential Operator ◽  
Cosine Transform ◽  
Invariant Differential

The goal of this paper is to describe the [Formula: see text]-cosine transform on functions on real Grassmannian [Formula: see text] in analytic terms as explicitly as possible. We show that for all but finitely many complex [Formula: see text] the [Formula: see text]-cosine transform is a composition of the [Formula: see text]-cosine transform with an explicitly written (though complicated) [Formula: see text]-invariant differential operator. For all exceptional values of [Formula: see text] except one, we interpret the [Formula: see text]-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value [Formula: see text], which is [Formula: see text], is still an open problem.

scholarly journals A CLASS OF STRONGLY SINGULAR RADON TRANSFORMS ON THE HEISENBERG GROUP

2007 ◽  
Vol 50 (2) ◽  
pp. 429-457 ◽  
Author(s):  
Neil Lyall
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
Singular Integrals ◽  
Radon Transforms ◽  
Laguerre Functions ◽  
Convolution Operators ◽  
Fourier Transform Methods ◽  
Group Fourier Transform ◽  
Asymptotic Forms ◽  
Singular Radon Transforms

AbstractWe primarily consider here the $L^2$ mapping properties of a class of strongly singular Radon transforms on the Heisenberg group $\mathbb{H}^n$; these are convolution operators on $\mathbb{H}^n$ with kernels of the form $M(z,t)=K(z)\delta_0(t)$, where $K$ is a strongly singular kernel on $\mathbb{C}^n$. Our results are obtained by using the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi.We also discuss the behaviour of related twisted strongly singular operators on $L^2(\mathbb{C}^n)$ and obtain results in this context independently of group Fourier transform methods. Key to this argument is a generalization of the results for classical strongly singular integrals on $L^2(\mathbb{R}^d)$.

On the kinetics of Hadamard-type fractional differential systems

2020 ◽  
Vol 23 (2) ◽  
pp. 553-570 ◽  
Author(s):  
Li Ma
Keyword(s):  
Differential Operator ◽  
Numerical Simulations ◽  
Type Inequality ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Kinetics Of ◽  
Theoretical Results

AbstractThis paper is devoted to the investigation of the kinetics of Hadamard-type fractional differential systems (HTFDSs) in two aspects. On one hand, the nonexistence of non-trivial periodic solutions for general HTFDSs, which are considered in some functional spaces, is proved and the corresponding eigenfunction of Hadamard-type fractional differential operator is also discussed. On the other hand, by the generalized Gronwall-type inequality, we estimate the bound of the Lyapunov exponents for HTFDSs. In addition, numerical simulations are addressed to verify the obtained theoretical results.

scholarly journals Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

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