%% ABSTRACT template %% 12th ICCA, Aveiro \documentclass[9pt]{amsart} % \usepackage{amsfonts,amsthm,amssymb,amsmath} \usepackage[scale={0.8,0.9}, a4paper, nofoot, bindingoffset=1cm]{geometry} \parindent0mm \parskip0mm \sloppy % \newtheorem*{theorem}{Theorem} % \newcommand{\ICCAabstract}[5]{{\bf #1}\par\medskip{\sc #2}\\{#4}\\{\tt #3} \par\bigskip {#5}\bigskip\bigskip} % \begin{document} \begin{twocolumn} % % Please do not change ANYTHING outside the following latex command and do NOT use % any hard font commands inside the command options (as \bf, \sf, \large etc.) except \empf{} if % you want to emphasise something. Please do not use labels and reference to formula numbers % in your abstract (as this will cause problems in the production process). You may use the % \begin{theorem} \end{theorem} environment to state theorems, use all font commands and % symbols provided by {amsfonts} and {amssymb} packages INSIDE formulae. % Please do NOT give references in or after your abstract. % % If your abstract exceeds 300 words in length, we will cut it after 300 words and replace % the remainder by \dots % \ICCAabstract% {%Title of Talk Uncertainty principle associated with the continuous Clifford wavelet Transform } {% Name Hicham BANOUH$^{\dagger}$ and Anouar Ben Mabrouk$^{\ddagger}$ } {% Email Address $^{\dagger}$hbanouh@usthb.dz \\ $^{\ddagger}$anouar.benmabrouk@fsm.rnu.tn } {% Address, NO LINE BREAKS $^{\dagger}$Laboratoire Systèmes Dynamiques, Faculty of Mathematics, USTHB, BP 32 Bab Ezzouar, 16111 - Algiers, Algeria.\\ $^{\ddagger}$Higher Institute of Applied Mathematics and Computer Science, University of Kairaouan, Street of Assad Ibn Alfourat, Kairaouan 3100, Tunisia. } {% Abstract % NOT TO EXCEED 300 WORDS The extension of wavelet analysis to Clifford framework is a relatively recent topic in applied mathematics. Conceptes of Clifford algebra and analysis are generalizations of associative non-commutative algebras, complex analysis, harmonic analysis as well as quaternion algebras.\\ In this work, we present an extension of the well known Heisenberg uncertainty principle to the case of some continuous wavelet transforms dedicated to the Clifford algebra $\mathbb{R}_{n}$. The main aim is to establish a lower bound of the product of variances of Clifford wavelet transform of a square integrable multivector-valued function and its Clifford-Fourier transform which means that they cannot both be estimated exactly, at the same time. Mathematically, the result may be summarized in the following theorem. \begin{theorem} Let $\psi$ be an admissible Clifford mother wavelet. Then for any square integrable multivector-valued function $f$, the following inequality holds, \begin{align*}{\displaystyle \left({\displaystyle {\displaystyle \underset{Spin(n)}{\int}\underset{\mathbb{R}^{+}}{\int}\underset{\mathbb{R}^{n}}{\int}\left|b_{k}T_{\psi}\left[f\right](a,\underline{b},s)\right|^{2}dV(\underline{b})\frac{da}{a^{n+1}}ds}}\right)^{\frac{1}{2}}}\\ \left(\underset{\mathbb{R}^{n}}{\int}\left|\xi_{k}\widehat{f}(\underline{\xi})\right|^{2}dV(\underline{b})\right)^{\frac{1}{2}}\\ \geq\frac{(2\pi)^{\frac{n}{2}}\sqrt{A_{\psi}}}{2}\left\Vert f\right\Vert _{2}^{2} \end{align*} for $k=1,2,\cdots,n$. \end{theorem}} The quantity $$A_{\psi}={\displaystyle (2\pi)^{n}\int_{\mathbb{R}^{n}}\frac{\widehat{\psi}(\underline{\xi})\left[\widehat{\psi}(\underline{\xi})\right]^{\dagger}}{|\underline{\xi}|^{n}}dV(\underline{\xi})<\infty.}$$ being the admissibility constant of the wavelet $\psi$ (a condition for the Clifford wavelet transform to be an invertible isometry), $Spin(n)$ the Spin group associated with $\mathbb{R}^{n}$ (a double cover of the $SO(n)$ group thus describing rotations) and $dV(\underline{b}),\frac{da}{a^{n+1}},ds$ are, respectively the measures on $\mathbb{R}^{n},\mathbb{R}^{+}$ and $Spin(n)$. We also present an improvement of the previous result in the form of a sharper lower bound.% \end{twocolumn} \end{document}