Construction of compactly supported affine frames

Since the publication, less than ten years ago, of Mallat’s paper on Multiresoltion Analysis [Ma], and Daubechies’ paper on the construction of smooth compactly supported refinable functions [D], wavelets had gained enormous popularity in mathematics and in the application domains. It is sufficient to note that there are currently more than 10,000 subscribers to the monthly Wavelet Digest. At the same time, the construction of concrete wavelet systems that are be useful for applications still remains a challenge. Specifically, simple and feasible constructions of orthonormal and bi-orthogonal systems of wavelets with small support, high smoothness and many symmetries is hard in more than one dimension (both tensor product methods, or the methods suggested in [RiS] and [JRS] yields wavelets with relatively large supports. In a series of recent articles [RS1-7] and [GR], a theory that changes the previous state-ofthe-art had been developed. That theory makes wavelet constructions simple and feasible, and it is the intent of the present article to provide a brief glance into it, with an emphasis on particular examples of univariate and multivariate constructs. We want to start with somewhat philosophical discussion: anyone who is familiar with wavelets knows that the simplest wavelet system is the Haar family. The Haar function is piecewise-constant, has a very small support, and the algorithms based on it are fast and simple. Had the Haar wavelet been found satisfactory, other wavelet constructions, together with the MRA framework, would have been superfluous. However, the frequency localization (read: the smoothness) of this wavelet is so bad, that improvements had been sought for at the outset. It is reasonable to argue that if piecewiseconstants are rejected, then continuous piecewise-linears are next in line: this is exactly the line of development in spline theory. Indeed, even before MRA was introduced, Battle [B], and Lemarié [L], constructed (independently) a piecewise-linear continuous spline with orthonormal dilated shifts (and knots at the half-integers only). Alas, that spline is of global support, and even its exponential decay at ∞ did not attract the masses, who deserted it in favor of Daubechies’ refinable functions and their bi-orthogonal off-springs (cf. [CDF]). The simplest function in Daubechies’ family [D] of refinable functions (i.e., that with support [0, 3]) is not piecewise-linear, but is related to piecewiselinears in some weak sense (its shifts reproduce all linear polynomials, just as the the shifts of the piecewise-linear hat function do); in any event, the question whether the corresponding wavelet is a ‘natural’ or ‘unnatural’ replacement for the Haar wavelet was not on the agenda anymore; rather, this wavelet is considered next in line the Haar’s because it is the continuous orthonormal wavelet with shortest support. Before we get to the main point of the present discussion, we need to introduce the notion of a tight frame. For that, we recall that, given any orthonormal system X for L2(IR ), we have