Generalizations and applications

Abstract

In our earlier discussion of “wavelet-like” bases in Hilbert space $\mathcal{H}$, we stressed the geometric point of view, which begins with a subspace V₀ and two operations. For standard wavelets in one variable, $\mathcal{H}$ will be the Hilbert space L²(ℝ), and a suitable “resolution subspace” V₀ will be chosen and assumed invariant under translation by the group of integers ℤ. In addition, it will be required that V₀ be invariant under some definite scaling operator, for example under “stretching” f → f(x/2). Thirdly, the traditional multiresolution (MRA) approach to L²(ℝ)-wavelets demands that the chosen subspace V₀ be singly generated, i.e., generated by a single function ϕ, the father function, i.e., the normalized L²-function which solves the scaling identity (see (1.3.1) in Chapter 1). As is known, it turns out that these demands for a subspace V₀ are rather stringent.

One cannot expect any serious understanding of what wavelet analysis means without a deep knowledge of the corresponding operator theory. —Yves Meyer