scholarly journals Optimal functions for a periodic uncertainty principle and multiresolution analysis

1999 ◽  
Vol 42 (2) ◽  
pp. 225-242 ◽  
Author(s):  
Jürgen Prestin ◽  
Ewald Quak
Keyword(s):  
Uncertainty Principle ◽  
Multiresolution Analysis ◽  
Theta Functions ◽  
Extremal Functions ◽  
Asymptotically Optimal ◽  
Time Frequency ◽  
Periodic Multiresolution Analysis ◽  
Optimal Functions

In this paper, it is shown that certain Theta functions are asymptotically optimal for the periodic time frequency uncertainty principle described by Breitenberger in [3]. These extremal functions give rise to a periodic multiresolution analysis where the corresponding wavelets also show similar localization properties.

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.

Local Price uncertainty principle and time-frequency localization operators for the Hankel–Stockwell transform

2020 ◽  
Vol 18 (06) ◽  
pp. 2050050
Author(s):  
Nadia Ben Hamadi ◽  
Zineb Hafirassou
Keyword(s):  
Uncertainty Principle ◽  
Price Uncertainty ◽  
Localization Operators ◽  
Von Neumann ◽  
Time Frequency ◽  
Neumann Class ◽  
Time Frequency Localization ◽  
Stockwell Transform

For the Hankel–Stockwell transform, the Price uncertainty principle is proved, we define the Localization operators and we study their boundedness and compactness. We also show that these operators belong to the so-called Schatten–von Neumann class.

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 and Extremal Functions for the Dunkl L2-Multiplier Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Fethi Soltani
Keyword(s):  
Uncertainty Principle ◽  
Tikhonov Regularization ◽  
Reproducing Kernels ◽  
Extremal Functions ◽  
Uncertainty Principles ◽  
Multiplier Operators

We study some class of Dunkl L2-multiplier operators; and related to these operators we establish the Heisenberg-Pauli-Weyl uncertainty principle and Donoho-Stark’s uncertainty principle. We give also an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev-Dunkl spaces.

SHAPIRO’S UNCERTAINTY PRINCIPLE IN THE DUNKL SETTING

2015 ◽  
Vol 92 (1) ◽  
pp. 98-110 ◽  
Author(s):  
SAIFALLAH GHOBBER
Keyword(s):  
Uncertainty Principle ◽  
Orthonormal Basis ◽  
Integral Transform ◽  
Reflection Group ◽  
Dunkl Transform ◽  
Time Frequency ◽  
Orthonormal Bases ◽  
Orthonormal Sequence ◽  
Finite Reflection Group ◽  
Uniformly Bounded

The Dunkl transform ${\mathcal{F}}_{k}$ is a generalisation of the usual Fourier transform to an integral transform invariant under a finite reflection group. The goal of this paper is to prove a strong uncertainty principle for orthonormal bases in the Dunkl setting which states that the product of generalised dispersions cannot be bounded for an orthonormal basis. Moreover, we obtain a quantitative version of Shapiro’s uncertainty principle on the time–frequency concentration of orthonormal sequences and show, in particular, that if the elements of an orthonormal sequence and their Dunkl transforms have uniformly bounded dispersions then the sequence is finite.

A Study of Matrix-Valued Trivariate Wavelet Packets Associated with an Orthogonal Matrix-Valued Trivariate Scaling Function

2010 ◽  
Vol 439-440 ◽  
pp. 1165-1170
Author(s):  
Jian Zhang ◽  
Shui Wang Guo
Keyword(s):  
Functional Analysis ◽  
Frequency Analysis ◽  
Multiresolution Analysis ◽  
Matrix Theory ◽  
Scaling Function ◽  
Orthogonal Matrix ◽  
Wavelet Packets ◽  
Riesz Bases ◽  
Analysis Method ◽  
Time Frequency

Wavelet analysis has become a developing branch of mathematics for over twenty years. In this paper, the notion of matrix-valued multiresolution analysis of space is introduced. A method for constructing biorthogonal matrix–valued trivariate wavelet packets is developed and their properties are discussed by means of time-frequency analysis method, matrix theory and functional analysis method. Three biorthogonality formulas concerning these wavelet packets are provided. Finally, new Riesz bases of space is obtained by constructing a series of subspaces of biorthogonal matrix-valued wavelet packets.

Sign in / Sign up

Export Citation Format

Share Document