Localisation fréquentielle des paquets d'ondelettes
Éric Séré · Revista Matemática Iberoamericana · 1995
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 .