scholarly journals A convolution type characterization forLp- multipliers for the Heisenberg group

2007 ◽  
Vol 5 (2) ◽  
pp. 175-182 ◽  
Author(s):  
R. Radha ◽  
A.K. Vijayarajan
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
Simple Proof ◽  
Convolution Type ◽  
Amenable Groups ◽  
Group Fourier Transform ◽  
The Fourier Transform

It is well known that ifmis anLp- multiplier for the Fourier transform onℝn(1<p<∞), then there exists a pseudomeasureσsuch thatTmf  =σ*f. A similar result is proved for the group Fourier transform on the Heisenberg groupHn. Though this result is already known in generality for amenable groups, a simple proof is provided in this paper.

scholarly journals Paley_ Wiener Theorems for The Fourier Transform depending on the Heisenberg group: مبرهنات بالي _ وينر لتحويل فورييه اعتماداً على زمرة هايزنبرغ

2020 ◽  
Vol 4 (3) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Fourier Transform ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Fourier Transforms ◽  
Mathematical Concepts ◽  
Weyl Transform ◽  
Wiener Theorem ◽  
Group Fourier Transform ◽  
The Fourier Transform ◽  
Compactly Supported Functions

  In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we discuss the representation theory of this group and the relationship between the representation theory of the Heisenberg group and the position and momentum operators and momentum operators relationship between the representation theory of the Heisenberg group and the position and momentum that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schrodinger picture. That is, all the representations we considered are realized in the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator-valued function, and other facts and properties. In our research, we depended on new formulas for some mathematical concepts such as Fourier Transform and Weyl transform. The main aim of our research is to introduce the Paley_ Wiener theorem for the Fourier transform on the Heisenberg group. We will show that the classical Paley_ Wiener theorem for the Euclidean Fourier transform characterizes compactly supported functions in terms of the behaviour of their Fourier transforms and Weyl transform. And we are interested in establishing results for the group Fourier transform and the Weyl transform.

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.

scholarly journals Revisiting the Fourier transform on the Heisenberg group

2014 ◽  
Vol 58 ◽  
pp. 47-63 ◽  
Author(s):  
R. Lakshmi Lavanya ◽  
S. Thangavelu
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
The Fourier Transform

scholarly journals A characterisation of the Fourier transform on the‎ ‎Heisenberg group

2012 ◽  
Vol 3 (1) ◽  
pp. 109-120 ◽  
Author(s):  
R‎. ‎Lakshmi Lavanya ◽  
S‎. ‎Thangavelu
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
The Fourier Transform

scholarly journals The role of Heisenberg group in Harmonic analysis: دور زمرة هايزنبرغ في التحليل التوافقي

2019 ◽  
Vol 3 (3) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Mathematical Concepts ◽  
Group Fourier Transform ◽  
The Relationship

In this paper we talk about Heisenberg group, the most know example from the lie groups. After that we discuss the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operatorsو and momentum operators.ors. ielationship between the representation theory of the Heisenberg group and the position and momen, that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schr dinger picture. That is, all the representations we considered are realized on the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator valued function, and other facts and properties. The main aim of our research is having the formula of Schr dinger Representation that connect physics with the Heisenberg group. Depending on this Representation we will study new formulas for some mathematical concepts such us Fourier Transform and  .

scholarly journals Uncertainty Principles for Heisenberg Motion Group

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Walid Amghar
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
Uncertainty Principles ◽  
Motion Group ◽  
The Fourier Transform

In this article, we will recall the main properties of the Fourier transform on the Heisenberg motion group G = ℍ n ⋊ K , where K = U n and ℍ n = ℂ n × ℝ denote the Heisenberg group. Then, we will present some uncertainty principles associated to this transform as Beurling, Hardy, and Gelfand-Shilov.

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 $L^2(\mathbb{R}^d)$.

scholarly journals Construction of Frames on the Heisenberg Groups

2020 ◽  
Vol 8 (1) ◽  
pp. 382-395
Author(s):  
Der-Chen Chang ◽  
Yongsheng Han ◽  
Xinfeng Wu
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
Operator Theory ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Identity Operator ◽  
Heisenberg Groups ◽  
Zygmund Operator ◽  
The Fourier Transform

Abstract In this paper, we present a construction of frames on the Heisenberg group without using the Fourier transform. Our methods are based on the Calderón-Zygmund operator theory and Coifman’s decomposition of the identity operator on the Heisenberg group. These methods are expected to be used in further studies of several complex variables.

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