Frequency localization of wavelet packets
Summary: 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 \(\widehat {w}_n\) as localization criterion, they showed that the average \(2^{- j} \sum_{n=0}^{2^j -1} |\widehat {w}_n |_{L^1}\) blows up as \(j\) goes to infinity. A natural problem is then to know which values of \(n\) create this blow-up in average. The present work gives an answer to this question, thanks to sharp estimates on \(|\widehat {w}_n |_{L^1}\) which depend on the dyadic expansion of \(n\), for several types of filters. Let us point out that the value of \(|\widehat {w}_n |_{L^1}\) is a weak localization criterion, which can only lead to a lower estimate on the variance of \(\widehat {w}_n\).
scientific article; zbMATH DE number 813825