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 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 Uncertainty principles invariant under the fractional Fourier transform

1991 ◽  
Vol 33 (2) ◽  
pp. 180-191 ◽  
Author(s):  
David Mustard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Intrinsic Property ◽  
Continuous Group ◽  
Uncertainty Principles ◽  
New Family ◽  
Special Cases ◽  
Fractional Fourier Transforms ◽  
The Fourier Transform

AbstractUncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

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

Numerical Synthesis of Binary Manipulator Workspaces Using the Fourier Transform on the Euclidean Motion Group

1998 ◽  
Author(s):  
Alexander Kyatkin ◽  
Gregory S. Chirikjian
Keyword(s):  
Fourier Transform ◽  
Analytical Form ◽  
Convolution Product ◽  
Generalized Convolution ◽  
Motion Group ◽  
Euclidean Motion Group ◽  
Linear Inverse Problems ◽  
Euclidean Motion ◽  
The Fourier Transform ◽  
Fitting Parameters

Abstract In this paper we apply the Fourier transform on the Euclidean motion group to solve problems in kinematic design of binary manipulators. We begin by reviewing how the workspace of a binary manipulator can be viewed as a function on the motion group, and how it can be generated as a generalized convolution product. We perform the convolution of manipulator densities, which results in the total workspace density of a manipulator composed of double the number of modules. We suggest an anzatz function which approximates the manipulator’s density in analytical form and has few free fitting parameters. Using the anzatz functions and Fourier methods on the motion group, linear and non-linear inverse problems (i. e. problems of finding the manipulator’s parameters which produce the total desired workspace density) are solved.

UNCERTAINTY PRINCIPLES FOR THE KONTOROVICH‐LEBEDEV TRANSFORM

2008 ◽  
Vol 13 (2) ◽  
pp. 289-302 ◽  
Author(s):  
Semyon B. Yakubovich
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principles ◽  
The Fourier Transform ◽  
Composition Properties

By using classical uncertainty principles for the Fourier transform and composition properties of the Kontorovich‐Lebedev transform, analogs of the Hardy, Beurling, Cowling‐Price, Gelfand‐Shilov and Donoho‐Stark theorems are obtained.

Some shaper uncertainty principles for multivector-valued functions

2016 ◽  
Vol 14 (06) ◽  
pp. 1650043
Author(s):  
Minggang Fei ◽  
Yubin Pan ◽  
Yuan Xu
Keyword(s):  
Fourier Transform ◽  
Clifford Algebra ◽  
Uncertainty Principle ◽  
Dunkl Transform ◽  
Uncertainty Principles ◽  
Heisenberg Uncertainty Principle ◽  
Clifford Fourier Transform ◽  
Heisenberg Uncertainty ◽  
The Fourier Transform ◽  
Adjoint Operators

The Heisenberg uncertainty principle and the uncertainty principle for self-adjoint operators have been known and applied for decades. In this paper, in the framework of Clifford algebra, we establish a stronger Heisenberg–Pauli–Wely type uncertainty principle for the Fourier transform of multivector-valued functions, which generalizes the recent results about uncertainty principles of Clifford–Fourier transform. At the end, we consider another stronger uncertainty principle for the Dunkl transform of multivector-valued functions.

scholarly journals The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1685
Author(s):  
Rui Jing ◽  
Bei Liu ◽  
Rui Li ◽  
Rui Liu
Keyword(s):  
Fourier Transform ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Transformation Operator ◽  
Local Analysis ◽  
Transform Domain ◽  
Uncertainty Principles ◽  
The Fourier Transform ◽  
Heisenberg's Uncertainty Principle ◽  
Fourier Transform Domain

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.

Synthesis of Binary Manipulators Using the Fourier Transform on the Euclidean Group

1999 ◽  
Vol 121 (1) ◽  
pp. 9-14 ◽  
Author(s):  
A. B. Kyatkin ◽  
G. S. Chirikjian
Keyword(s):  
Fourier Transform ◽  
Inverse Problems ◽  
Analytical Form ◽  
Convolution Product ◽  
Euclidean Group ◽  
Motion Group ◽  
Euclidean Motion Group ◽  
Linear Inverse Problems ◽  
Euclidean Motion ◽  
The Fourier Transform

In this paper we apply the Fourier transform on the Euclidean motion group to solve problems in kinematic design of binary manipulators. In recent papers it has been shown that the workspace of a binary manipulator can be viewed as a function on the motion group, and it can be generated as a generalized convolution product. The new contribution of this paper is the numerical solution of mathematical inverse problems associated with the design of binary manipulators. We suggest an anzatz function which approximates the manipulator’s density in analytical form and has few free fitting parameters. Using the anzatz functions and Fourier methods on the motion group, linear and non-linear inverse problems (i.e., problems of finding the manipulator’s parameters which produce the total desired workspace density) are solved.

Sign in / Sign up

Export Citation Format

Share Document