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 $wn$ 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 $\\hat{w}\_n$.
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