scholarly journals Variations on a theorem of Cowling and Price with applications to nilpotent Lie groups

2007 ◽  
Vol 82 (1) ◽  
pp. 11-27 ◽  
Author(s):  
S. Parui ◽  
S. Thangavelu
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Uncertainty Principle ◽  
Fourier Transforms ◽  
Nilpotent Lie Groups

AbstractIn this paper we prove a new version of the Cowling-Price theorem for Fourier transforms on Rn. Using this we formulate and prove an uncertainty principle for operators. This leads to an analogue of the Cowling-Price theorem for nilpotent Lie groups. We also prove an exact analogue of the Cowling-Price theorem for the Heisenberg group.

scholarly journals Qualitative uncertainty principle for the Gabor transform on certain locally compact groups

2018 ◽  
Vol 9 (3) ◽  
pp. 205-220
Author(s):  
Jyoti Sharma ◽  
Ajay Kumar
Keyword(s):  
Lie Groups ◽  
Uncertainty Principle ◽  
Discrete Group ◽  
Locally Compact Groups ◽  
Gabor Transform ◽  
Compact Groups ◽  
Nilpotent Lie Groups ◽  
Locally Compact ◽  
Qualitative Uncertainty ◽  
Low Dimensional

Abstract Several classes of locally compact groups have been shown to possess a qualitative uncertainty principle for the Gabor transform. These include Moore groups, the Heisenberg group {\mathbb{H}_{n}} , the group {\mathbb{H}_{n}\times D} (where D is a discrete group) and other low-dimensional nilpotent Lie groups.

scholarly journals Cowling Price Theorem for Low Dimensional Nilpotent Lie Groups

2010 ◽  
Vol 27 (1-2) ◽  
pp. 49-52
Author(s):  
C. R. Bhatta
Keyword(s):  
Lie Groups ◽  
Uncertainty Principle ◽  
Classical Result ◽  
Nilpotent Lie Groups ◽  
Low Dimensional

We extend an uncertainty principle due to Cowling and Price to low dimensional Nilpotent Lie groups G4. The uncertainty principle is age neralization of a classical result due to Hardy.

scholarly journals Studying in Lee groups and most important examples of it (Heisenberg group): دراسة في زمر لي وأهم الأمثلة عنها (زمرة هايزنبرغ)

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Number Theory ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
General Definition ◽  
Nilpotent Lie Groups ◽  
Basic Facts

In this research, we present some basic facts about Lie algebra and Lie groups. We shall require only elementary facts about the general definition and knowledge of a few of the more basic groups, such as Euclidean groups. Then we introduce the Heisenberg group which is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis.

scholarly journals Uncertainty relations on nilpotent Lie groups

2017 ◽  
Vol 473 (2201) ◽  
pp. 20170082 ◽  
Author(s):  
Michael Ruzhansky ◽  
Durvudkhan Suragan
Keyword(s):  
Quantum Mechanics ◽  
Heisenberg Group ◽  
Lie Groups ◽  
Coulomb Potential ◽  
Nilpotent Lie Groups ◽  
Uncertainty Relations ◽  
Homogeneous Groups ◽  
Potential Operators ◽  
Nirenberg Inequality ◽  
Uncertainty Inequalities

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic R n , and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.

scholarly journals Hardy Uncertainty Principle for Low Dimensional Nilpotent Lie Groups G4 (III)

1970 ◽  
Vol 6 (1) ◽  
pp. 89-95
Author(s):  
CR Bhatta
Keyword(s):  
Fourier Transform ◽  
Lie Groups ◽  
Uncertainty Principle ◽  
Nilpotent Lie Groups ◽  
Hardy’S Theorem ◽  
Keywords And Phrases ◽  
Engineering And Technology ◽  
The Fourier Transform ◽  
Low Dimensional

An uncertainty principle due to Hardy for Fourier transform pairs on ℜ says that if thefunction f is "very rapidly decreasing", then the Fourier transform can not also be"very rapidly decreasing" unless f is identically zero. In this paper we state and provean analogue of Hardy's theorem for low dimensional nilpotent Lie groups G4.Keywords and phrases: Uncertainty principle; Fourier transform pairs; veryrapidly decreasing; Nilpotent Lie groups.DOI: 10.3126/kuset.v6i1.3315 Kathmandu University Journal of Science, Engineering and Technology Vol.6(1) 2010, pp89-95

scholarly journals An uncertainty principle like Hardy's theorem for nilpotent Lie groups

2004 ◽  
Vol 77 (1) ◽  
pp. 47-54 ◽  
Author(s):  
Ajay Kumar ◽  
Chet Raj Bhatta
Keyword(s):  
Lie Groups ◽  
Uncertainty Principle ◽  
Classical Result ◽  
Nilpotent Lie Groups ◽  
Heisenberg Groups ◽  
Hardy’S Theorem

AbstractWe extend an uncertainty principle due to Cowling and Price to threadlike nilpotent Lie groups. This uncertainty principle is a generalization of a classical result due to Hardy. We are thus extending earlier work on Rnand Heisenberg groups.

scholarly journals The Sub- Laplacian on The Heisenberg Group and Itʼs Spectral Properties: مؤثر لابلاس الجزئي على زمرة هايزنبرغ وخصائصه الطيفية

2020 ◽  
Vol 6 (1) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Number Theory ◽  
Heisenberg Group ◽  
Lie Groups ◽  
Spectral Properties ◽  
Complete Analysis ◽  
Nilpotent Lie Groups ◽  
One Dimensional ◽  
Underlying Space ◽  
Adjoint Extension ◽  
The One

In this paper we talk about the spectral theory of the sub-Laplacian on the Heisenberg group. Then we give a complete analysis of the spectrum of the unique self- adjoint extension of this sub-Laplacian on the one-dimensional Heisenberg group. The Heisenberg group is the most known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis. The results in this paper are valid for the sub-Laplacian on the n-dimensional Heisenberg group, in which the underlying space is, but we have chosen to present the results for the one-dimensional Heisenberg group ℍ for the sake of simplicity and transparency.

scholarly journals Low-dimensional Nilpotent Lie Groups G4

1970 ◽  
Vol 10 ◽  
pp. 155-159
Author(s):  
Chet Raj Bhatta
Keyword(s):  
Fourier Transform ◽  
Key Words ◽  
Lie Groups ◽  
Uncertainty Principle ◽  
Science And Technology ◽  
Nilpotent Lie Groups ◽  
The Fourier Transform ◽  
Low Dimensional

An uncertainty principle due to Hardy for Fourier transform pairs on R says that if the function f is "very rapidly decreasing" then the Fourier transform cannot also be "very rapidly decreasing unless f is indentically zero." In this paper we study the relevant data for G4 and state and prove an analogue of Hardy theorem for low-dimensional nilpotent Lie groups G4.Key words: Fourier transform; Uncertainity principle; Nilpotent Lie groupsDOI: 10.3126/njst.v10i0.2951Nepal Journal of Science and Technology Vol. 10, 2009 Page: 155-159 

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