Sharper uncertainty principle and Paley–Wiener theorem for the Dunkl transform

2021 ◽  
Author(s):  
Minggang Fei ◽  
Qian Wang ◽  
Lan Yang
Keyword(s):  
Uncertainty Principle ◽  
Dunkl Transform ◽  
Wiener Theorem

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.

scholarly journals The Range of the Spectral Projection Associated with the Dunkl Laplacian

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Salem Ben Said ◽  
Hatem Mejjaoli
Keyword(s):  
Spectral Projection ◽  
Closed Ball ◽  
Difference Operators ◽  
Dunkl Transform ◽  
Wiener Theorem ◽  
Dunkl Laplacian

For s∈ℝ, denote by Pksf the “projection” of a function f in Dℝd into the eigenspaces of the Dunkl Laplacian Δk corresponding to the eigenvalue −s2. The parameter k comes from Dunkl’s theory of differential-difference operators. We shall characterize the range of Pks on the space of functions f∈Dℝd supported inside the closed ball BO,R¯. As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform.

scholarly journals Sharp Pitt inequality and logarithmic uncertainty principle for Dunkl transform in L2

2016 ◽  
Vol 202 ◽  
pp. 109-118 ◽  
Author(s):  
D.V. Gorbachev ◽  
V.I. Ivanov ◽  
S.Yu. Tikhonov
Keyword(s):  
Uncertainty Principle ◽  
Dunkl Transform

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 An uncertainty principle for the Dunkl transform

1999 ◽  
Vol 59 (3) ◽  
pp. 353-360 ◽  
Author(s):  
Margit Rösler
Keyword(s):  
Uncertainty Principle ◽  
Hermite Functions ◽  
Dunkl Transform

This note presents an analogue of the classical Heisenberg-Weyl uncertainty principle for the Dunkl transform on ℝN. Its proof is based on expansions with respect to generalised Hermite functions.

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