scholarly journals Helgason–Gabor–Fourier transform and uncertainty principles

2020 ◽  
pp. 2050056
Author(s):  
Mustapha Boujeddaine ◽  
Mohammed El Kassimi ◽  
Saïd Fahlaoui
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Plancherel Formula ◽  
Riemannian Symmetric Space ◽  
Uncertainty Principles ◽  
Reconstruction Formula ◽  
Time Frequency ◽  
Local Uncertainty Principle ◽  
Local Uncertainty ◽  
Parseval Formula

Windowing a Fourier transform is a useful tool, which gives us the similarity between the signal and time frequency signal, and it allows to get sense when/where certain frequencies occur in the input signal, this method was introduced by Dennis Gabor. In this paper, we generalize the classical Gabor–Fourier transform (GFT) to the Riemannian symmetric space calling it the Helgason–Gabor–Fourier transform (HGFT). We prove several important properties of HGFT like the reconstruction formula, the Plancherel formula and Parseval formula. Finally, we establish some local uncertainty principle such as Benedicks-type uncertainty principle.

scholarly journals Uncertainty Principles for the Dunkl-Wigner Transforms

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Fethi Soltani
Keyword(s):  
Uncertainty Principle ◽  
Uncertainty Principles ◽  
Wigner Transform ◽  
Heisenberg Type ◽  
Local Uncertainty Principle ◽  
Local Uncertainty

We prove a version of Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitude s>0; and we deduce a local uncertainty principle for this transform.

scholarly journals A variety of uncertainty principles for the Hankel-Stockwell transform

2021 ◽  
Vol 5 (1) ◽  
pp. 22-34
Author(s):  
Khaled Hleili ◽  
◽  
Keyword(s):  
Uncertainty Principle ◽  
Uncertainty Principles ◽  
Local Uncertainty Principle ◽  
Local Uncertainty ◽  
Pitt’S Inequality ◽  
Stockwell Transform

In this work, we establish \(L^p\) local uncertainty principle for the Hankel-Stockwell transform and we deduce \(L^p\) version of Heisenberg-Pauli-Weyl uncertainty principle. Next, By combining these principles and the techniques of Donoho-Stark we present uncertainty principles of concentration type in the \(L^p\) theory, when \(1< p\leqslant2\). Finally, Pitt's inequality and Beckner's uncertainty principle are proved for this transform.

Certifying Einstein-Podolsky-Rosen steering via the local uncertainty principle

2016 ◽  
Vol 93 (1) ◽  
Author(s):  
Yi-Zheng Zhen ◽  
Yu-Lin Zheng ◽  
Wen-Fei Cao ◽  
Li Li ◽  
Zeng-Bing Chen ◽  
...  
Keyword(s):  
Uncertainty Principle ◽  
Local Uncertainty Principle ◽  
Local Uncertainty ◽  
Einstein Podolsky Rosen

scholarly journals Global and local uncertainty principles for signals on graphs

2018 ◽  
Vol 7 ◽  
Author(s):  
Nathanael Perraudin ◽  
Benjamin Ricaud ◽  
David I Shuman ◽  
Pierre Vandergheynst
Keyword(s):  
Local Structure ◽  
Small Region ◽  
Uncertainty Principles ◽  
Missing Information ◽  
Spectral Filter ◽  
Time Frequency ◽  
Local Uncertainty ◽  
Linear Signal ◽  
Global And Local ◽  
Global Uncertainty

Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design, lead to algorithms for reconstructing missing information via sparse representations, and yield new graph analysis tools. While previous work has focused on generalizing notions of spreads of graph signals in the vertex and graph spectral domains, our approach generalizes the methods of Lieb in order to develop uncertainty principles that provide limits on the concentration of the analysis coefficients of any graph signal under a dictionary transform. One challenge we highlight is that the local structure in a small region of an inhomogeneous graph can drastically affect the uncertainty bounds, limiting the information provided by global uncertainty principles. Accordingly, we suggest new notions of locality, and develop local uncertainty principles that bound the concentration of the analysis coefficients of each atom of a localized graph spectral filter frame in terms of quantities that depend on the local structure of the graph around the atom's center vertex. Finally, we demonstrate how our proposed local uncertainty measures can improve the random sampling of graph signals.

A Local Uncertainty Principle

1984 ◽  
Vol 15 (5) ◽  
pp. 988-995
Author(s):  
John J. Benedetto
Keyword(s):  
Uncertainty Principle ◽  
Local Uncertainty Principle ◽  
Local Uncertainty

Dispersion’s Uncertainty Principles Associated with the Directional Short-Time Fourier Transform

2020 ◽  
Vol 57 (4) ◽  
pp. 508-540
Author(s):  
Siwar Hkimi ◽  
Hatem Mejjaoli ◽  
Slim Omri
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Price Uncertainty ◽  
Uncertainty Principles ◽  
Short Time Fourier Transform ◽  
Short Time

We introduce the directional short-time Fourier transform for which we prove a new Plancherel’s formula. We also prove for this transform several uncertainty principles as Heisenberg inequalities, logarithmic uncertainty principle, Faris–Price uncertainty principles and Donoho–Stark’s uncertainty principles.

scholarly journals Localization Operators and an Uncertainty Principle for the Discrete Short Time Fourier Transform

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Carmen Fernández ◽  
Antonio Galbis ◽  
Josep Martínez
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Extremal Functions ◽  
Short Time Fourier Transform ◽  
Localization Operators ◽  
Time Frequency ◽  
Short Time ◽  
Frequency Basis ◽  
Discrete Setting

Localization operators in the discrete setting are used to obtain information on a signalffrom the knowledge on the support of its short time Fourier transform. In particular, the extremal functions of the uncertainty principle for the discrete short time Fourier transform are characterized and their connection with functions that generate a time-frequency basis is studied.

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.

scholarly journals On the Quaternionic Short-Time Fourier and Segal–Bargmann Transforms

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
Antonino De Martino ◽  
Kamal Diki
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Reproducing Kernel ◽  
Short Time Fourier Transform ◽  
Reconstruction Formula ◽  
Bargmann Transform ◽  
One Dimensional ◽  
Basic Properties ◽  
Bargmann Transforms ◽  
Short Time

AbstractIn this paper, we study a special one-dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal–Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb’s uncertainty principle. We provide also the reproducing kernel associated with the Gabor space considered in this setting.

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