Some uncertainty inequalities for the continuous wavelet transform
type: Item
instance of: scholarly article
title: Some uncertainty inequalities for the continuous wavelet transform
zbMATH Open document ID: 1477.42036
author: Fatima Alhajji
author: Saifallah Ghobber
published in: Journal of Pseudo-Differential Operators and Applications
publication date: 2021-08-12T00:00:00Z
review text: In this paper, the authors prove some uncertainty inequalities for the continuous wavelet transform in the multidimensional setting. A nonzero function \(\psi \in L^2(\mathbb{R}^d, \mu_d),\) where \(\mu_d\) is the normalized Lebesgue measure on \(\mathbb{R}^d,\) is said to be a mother wavelet (or an admissible wavelet) if it satisfies the following admissibility condition: \[ 0 < c_{\psi} := \int_0^{\infty} \frac 1a |\widehat{\psi}(a \xi)|^2 \ d a < \infty, \quad \forall \xi \in \mathbb{R}^d \setminus \{0\} : |\xi|=1, \] where \(| \cdot |\) is the usual norm on \(\mathbb{R}^d,\) and \(\widehat{\psi}\) represents the Fourier transform. The daughter wavelet of \(\psi\) is defined by \[ \psi_{a,b}(x) = \frac{1}{\sqrt{c_{\psi}}} \frac{1}{a^{d/2}}\psi\left(\frac{x - a}{b}\right), \quad (a, b) \in \mathbb{U} = (0, \infty) \times \mathbb{R}^d. \] Let \(f \in L^2(\mathbb{R}^d, \mu_d),\) and let \(\mathcal{W}_{\psi} f\) be its wavelet transform. Consider the left invariant Haar measure \(d \mu_{\textrm{aff}}(a,b) = a^{-(d+1)} da \ d_{\mu_d}(b).\) There are three main results in this paper: \par 1) For all \(f \in L^2(\mathbb{R}^d, \mu_d),\) it is shown that \[ \| \ |b| \mathcal{W}_{\psi} f \ \|_{2, \mu_{\textrm{aff}}} \ \| \ |\xi| \widehat{f}\ \|_{2, \mu_{d}} \geq \frac d2\|f\|^2_{2, \mu_{d}}. \] For \(s,\) \(\beta \geq 1\), this gives rise to \[ \| \ |b|^s \mathcal{W}_{\psi} f \ \|^{\beta}_{2, \mu_{\textrm{aff}}} \ \| \ |\xi|^{\beta} \widehat{f}\ \|^{s}_{2, \mu_{d}} \geq \left(\frac d2 \right)^{s \beta} \|f\|^{s + \beta}_{2, \mu_{d}} \] which can be looked upon as a Heisenberg-type uncertainty inequality for the continuous wavelet transform in the multidimensional setting. One can compare the concentration of \(f\) in frequency (given by \(\xi\)) with the concentration of its wavelet transform \(\mathcal{W}_{\psi} f\) in position (given by \(b\)). \par 2) Given a set \(A\), \(PW_{A} = \{f \in L^2(\mathbb{R}^d, \mu_d): \textrm{supp} \widehat{f} \subset A\}\) is the Paley-Wie…
reviewed by: Somantika Datta
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 42C25
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 7381172
zbMATH Keywords: wavelet transform
zbMATH Keywords: affine group
zbMATH Keywords: Heisenberg uncertainty inequality
zbMATH Keywords: Pitt's inequality
zbMATH Keywords: Beckner's inequality
zbMATH Keywords: uncertainty principle
zbMATH Keywords: Sobolev's inequality
MaRDI profile type: Publication
full work available at URL: s11868-021-00388-w
OpenAlex ID: W3168520509
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DOI: 10.1007/S11868-021-00388-W
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description: scientific article; zbMATH DE number 7381172