Size properties of wavelet-packets. (English) Zbl 0822.42019

Coifman, R. R.; Meyer, Y.; Wickerhauser, V.

Ruskai, Mary Beth (ed.) et al., Wavelets and their applications. Boston, MA etc.: Jones and Bartlett Publishers. 453-470 (1992).

This paper begins with a very clear description, in Sections 2 through 5, of how wavelet packets are derived from orthonormal bases. Sections 6 and 7 are devoted to a study of the growth properties of the \(L^ \infty\) and \(L^ p\) norms of wavelet packets, with the goal of understanding the frequency localization of basic wavelet packets, for which lower bounds are given by the asymptotic behavior of the \(L^ \infty\) norms of wavelet packets. In the introduction, the authors observe that their results imply that basic wavelet packet functions are not as sharply localized in frequency as one might hope.
In Section 8, the authors investigate an extended notion of wavelet packets, subsets of which are indexed by a covering set (possibly infinite) of dyadic subintervals on an interval. Theorem 6 shows how to construct orthonormal bases of \(L^ 2(\mathbb{R})\) using generalized wavelet packets. In the conclusion, the authors show that Theorem 6 does not completely characterize all possible wavelet-packet orthonormal bases, some of which may correspond to dyadic subintervals which do not cover all but an exceptional denumerable subset of an interval.
For the entire collection see [Zbl 0782.00087].

Reviewer: C.Pfeffer Johnston (Boca Raton)

MSC:

42C40

Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

wavelet packets; frequency localization; subsets; dyadic subintervals