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.

Symmetric Wilson systems

2019 ◽  
Vol 17 (05) ◽  
pp. 1950036
Author(s):  
Piotr Wojdyłło
Keyword(s):  
Normal Form ◽  
Orthonormal Basis ◽  
Tight Frame ◽  
Generator Matrix ◽  
Frequency Shifts ◽  
Time Frequency ◽  
Orthonormal Bases ◽  
Full Characterization ◽  
Hermite Normal Form

The Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé using combinations of elements of Gabor tight frame with redundancy 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Then, Kutyniok and Strohmer generalized the notion of Wilson system to the lattices whose generator matrix is in Hermite normal form.In the present publication, we introduce Wilson systems where the time-frequency shifts are combined symmetrically with respect to the origin. Using the arguments from previous papers that work also in this case, we give a full characterization of atoms for which the obtained Wilson systems are orthonormal bases in [Formula: see text] and we generalize these results to different sign sequences and to symplectic lattices of density [Formula: see text].

scholarly journals Uncertainty Principles on Weighted Spheres, Balls, and Simplexes

2016 ◽  
Vol 59 (01) ◽  
pp. 62-72
Author(s):  
Han Feng
Keyword(s):  
Weight Function ◽  
Uncertainty Principle ◽  
Unit Sphere ◽  
Reflection Group ◽  
Beltrami Operator ◽  
Uncertainty Principles ◽  
Finite Reflection Group ◽  
Heisenberg Inequality ◽  
Full Analogy

Abstract This paper studies the uncertainty principle for spherical h-harmonic expansions on the unit sphere of ℝ d associated with a weight function invariant under a general finite reflection group, which is in full analogy with the classical Heisenberg inequality. Our proof is motivated by a new decomposition of the Dunkl–Laplace–Beltrami operator on the weighted sphere.

Orthonormal basis of wavelets with customizable frequency bands

2016 ◽  
Vol 14 (06) ◽  
pp. 1650050 ◽  
Author(s):  
Hiroshi Toda ◽  
Zhong Zhang
Keyword(s):  
Frequency Domain ◽  
Infinite Number ◽  
Orthonormal Basis ◽  
The Other ◽  
Just Intonation ◽  
Frequency Bands ◽  
Orthonormal Bases ◽  
Major Scale ◽  
Dilation Factor ◽  
New Type

We already proved the existence of an orthonormal basis of wavelets having an irrational dilation factor with an infinite number of wavelet shapes, and based on its theory, we proposed an orthonormal basis of wavelets with an arbitrary real dilation factor. In this paper, with the development of these fundamentals, we propose a new type of orthonormal basis of wavelets with customizable frequency bands. Its frequency bands can be freely designed with arbitrary bounds in the frequency domain. For example, we show two types of orthonormal bases of wavelets. One of them has an irrational dilation factor, and the other is designed based on the major scale in just intonation.

scholarly journals Twisted Invariant Theory for Reflection Groups

2006 ◽  
Vol 182 ◽  
pp. 135-170 ◽  
Author(s):  
C. Bonnafé ◽  
G. I. Lehrer ◽  
J. Michel
Keyword(s):  
Invariant Theory ◽  
Regular Element ◽  
Central Element ◽  
Reflection Group ◽  
General Criterion ◽  
Coordinate Ring ◽  
Reflection Groups ◽  
Complex Vector ◽  
Finite Reflection Group ◽  
Module Structures

AbstractLet G be a finite reflection group acting in a complex vector space V = ℂr, whose coordinate ring will be denoted by S. Any element γ ∈ GL(V) which normalises G acts on the ring SG of G-invariants. We attach invariants of the coset Gγ to this action, and show that if G′ is a parabolic subgroup of G, also normalised by γ, the invariants attaching to G′γ are essentially the same as those of Gγ. Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of G and G′ and secondly, we give a general criterion for an element of Gγ to be regular (in Springer’s sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of G which are themselves reflection groups.

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 Distance to the convex hull of an orbit under the action of a compact Lie group

1999 ◽  
Vol 66 (3) ◽  
pp. 331-357 ◽  
Author(s):  
Randall R. Holmes ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Analogous Result ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Adjoint Action ◽  
Reflection Group ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Reflection Group

AbstractFor a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

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 Note on Cubature Formulae and Designs Obtained from Group Orbits

2012 ◽  
Vol 64 (6) ◽  
pp. 1359-1377 ◽  
Author(s):  
Hiroshi Nozaki ◽  
Masanori Sawa
Keyword(s):  
Cubature Formula ◽  
Sufficient Conditions ◽  
Reflection Group ◽  
Alternative Proof ◽  
Invariant Polynomials ◽  
Cubature Formulas ◽  
Finite Reflection Group ◽  
Necessary And Sufficient ◽  
Cubature Formulae ◽  
Invariant Cubature Formulas

Abstract In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t . In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B, to other finite reflection groups.

The Presentation of Biorthogonal Non-Tensor Trivariate Wavelet Wraps in Besov Space

2012 ◽  
Vol 461 ◽  
pp. 738-742
Author(s):  
De Lin Hua
Keyword(s):  
Besov Space ◽  
Frequency Analysis ◽  
Iteration Method ◽  
Filter Bank ◽  
Matrix Theory ◽  
Analysis Method ◽  
Analogy Method ◽  
Time Frequency ◽  
Orthonormal Bases ◽  
Vector Valued

In this paper, the concept of orthogonal non-tensor bivariate wavelet packs, which is the generalization of orthogonal univariate wavelet packs, is pro -posed by virtue of analogy method and iteration method. Their orthogonality property is investigated by using time-frequency analysis method and variable se-paration approach. Three orthogonality formulas regarding these wavelet wraps are established. Moreover, it is shown how to draw new orthonormal bases of space from these wavelet wraps. A procedure for designing a class of orthogonal vector-valued finitely supported wavelet functions is proposed by virtue of filter bank theory and matrix theory.

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