# Localisation fréquentielle des paquets d'ondelettes

### Éric Séré

Université de Paris Dauphine, France

## 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 $2^{–j} \sum^{2^j–1}_{n=0} \|\hat{w}_n\|_{L^1}$ 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 $\|\hat{w}_n\|_{L^1}$ which depend on the dyadic expansion of $n$ for several types of filters. Let us point out that the value of $\|\hat{w}_n\|_{L^1}$ is a weak localization criterion, which can only lead to a lower estimate on the variance of $\hat{w}_n$.

## Cite this article

Éric Séré, Localisation fréquentielle des paquets d'ondelettes. Rev. Mat. Iberoam. 11 (1995), no. 2, pp. 334–354

DOI 10.4171/RMI/175