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Heisenberg-Pauli-Weyl uncertainty principles for the fractional Dunkl transform on the real line

Sunit Ghosh, Younis Ahmad Bhat, Jitendriya Swain

Abstract

The aim of the paper is two-fold. First, we provide an explicit form of the functions for which equality holds for the uncertainty inequalities studied in \cite{Fei}. Second, we establish an $L^p$-type Heisenberg-Pauli-Weyl uncertainty principle for the fractional Dunkl transform, with $1 \leq p \leq 2$. For the case $p = 2$, we further derive a sharper uncertainty principle for the fractional Dunkl transform. Furthermore, we derive conditions leading to equality in both the uncertainty principles obtained.

Heisenberg-Pauli-Weyl uncertainty principles for the fractional Dunkl transform on the real line

Abstract

The aim of the paper is two-fold. First, we provide an explicit form of the functions for which equality holds for the uncertainty inequalities studied in \cite{Fei}. Second, we establish an -type Heisenberg-Pauli-Weyl uncertainty principle for the fractional Dunkl transform, with . For the case , we further derive a sharper uncertainty principle for the fractional Dunkl transform. Furthermore, we derive conditions leading to equality in both the uncertainty principles obtained.

Paper Structure

This paper contains 6 sections, 9 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dunkl transform and the Dunkl operator
  4. Fractional Dunkl transform
  5. Covariance and absolute covariance
  6. HPWUP for the fractional Dunkl Transform:

Key Result

Theorem 1.1

Rosler. Let $xf(x)\in L^2_{\mu}(\mathbb{R})~ \text{and}~~xD_{\mu}(f)(x)\in L^2_{\mu}(\mathbb{R})$ with $\|f\|_{\mu,2}=1$. Then equality holds in (Rosler) if and only if $f$ has the form $f(x)=de^{-\frac{1}{2\zeta}x^2} E_{\mu}(bx)$, with $b\in\mathbb{C}$ and $\zeta>0$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 3.1
  • ...and 3 more