Best-adapted wavelet packet bases
Mladen Victor Wickerhauser · Proceedings of symposia in applied mathematics · 1993
. This paper is a review of the construction of orthogonal wavelet packets, using the quadrature mirror filter algorithm slightly generalized to the case of p 2 wavelets and scaling functions. It is part of the AMS short course on "Wavelets and Applications" held in San Antonio, 11-12 January 1993. Introduction. We begin with a classical reproducing formula for functions f 2 H 2 , the Hardy space of square-integrable functions whose Fourier transforms vanish on the negative half-line: If c = 2ß Z 1 0 j /(¸)j 2 j¸j d¸ ! 1 and T f (a; b) = Z R f(x) ¯ /(ax + b) dx; then f(x) = 1 c ZZ R\\ThetaR + Wf(a; b)/(ax + b) da db This formula was studied by Calder'on in the 60's and revived by Grossmann and Morlet in their 1984 paper [GM]. A function / satisfying the admissibility condition c ! 1 is called a "wavelet," and the map f 7! T f is called the (continuous) wavelet transform. The discrete (dyadic) wavelet transform transform is the restriction f 7! fT f (2 j ; k), j; ...