Generalizations and applications

Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 234)

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” ff(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.

Keywords

Hilbert Space Wavelet Packet Wavelet Filter Daubechies Wavelet Dyadic Wavelet
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