Localisation fréquentielle des paquets d'ondelettes

Abstract

Orthonormal bases of wavelet packets constitute a powerful tool in signal compression. It has been proved by Coifman, Meyer and Wickerhauser that "many" wavelet packets $w_n$ suffer a lack of frequency localization. Using the $L^1$-norm of the Fourier transform ${\hat{w}}_n$ as localization criterion, they showed that the average $2^{–j}{\sum }_{n=0}^{2^j–1}\Vert {\hat{w}}_n{\Vert }_{L^1}$ blows up as $j$ goes to infinity. A natural problem is then to know which values of $n$ create this blowup in average. The present work gives an answer to this question thanks to sharp estimates on $\Vert {\hat{w}}_n{\Vert }_{L^1}$ which depend on the dyadic expansion of $n$ for several types of filters. Let us point out that the value of $\Vert {\hat{w}}_n{\Vert }_{L^1}$ is a weak localization criterion, which can only lead to a lower estimate on the variance of ${\hat{w}}_n$.