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.

Advances in Analysis

2014 ◽  
Author(s):  
Charles Fefferman ◽  
Alexandru D. Ionescu ◽  
D.H. Phong ◽  
Stephen Wainger
Keyword(s):  
Fourier Transform ◽  
Representation Theory ◽  
Fourier Analysis ◽  
Interpolation Theorem ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Restriction Theorem ◽  
Several Variables ◽  
Hp Spaces ◽  
The Fourier Transform

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze–Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein's contributions to harmonic analysis and related topics, this book gathers papers from internationally renowned mathematicians, many of whom have been Stein's students. The book also includes expository papers on Stein's work and its influence.

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

Heat kernels and Green functions of sub-Laplacians on Heisenberg groups with multi-dimensional center

2018 ◽  
Vol 13 (4) ◽  
pp. 38
Author(s):  
Shahla Molahajloo ◽  
M.W. Wong
Keyword(s):  
Fourier Transform ◽  
Green Function ◽  
Heisenberg Group ◽  
Inverse Fourier Transform ◽  
Heat Kernels ◽  
Hermite Functions ◽  
Green Functions ◽  
Heisenberg Groups ◽  
Explicit Formulas ◽  
Special Hermite Functions

We compute the sub-Laplacian on the Heisenberg group with multi-dimensional center. By taking the inverse Fourier transform with respect to the center, we get the parametrized twisted Laplacians. Then by means of the special Hermite functions, we find the eigenfunctions and the eigenvalues of the twisted Laplacians. The explicit formulas for the heat kernels and Green functions of the twisted Laplacians can then be obtained. Then we give an explicit formula for the heat kernal and Green function of the sub-Laplacian on the Heisenberg group with multi-dimensional center.

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 On a space of entire functions rapidly decreasing on Rn and its Fourier transform

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Il’dar Kh. Musin
Keyword(s):  
Fourier Transform ◽  
Entire Functions ◽  
Type Theorem ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Weight Functions ◽  
Partial Derivatives ◽  
Wiener Type ◽  
Equivalent Description

AbstractA space of entire functions of several complex variables rapidly decreasing on Rn and such that their growth along iRn is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on Rn and a Paley-Wiener type theorem are obtained.

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 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.

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