# Generalizations and applications

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## 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 *V*_{0} and two operations. For standard wavelets in one variable, \(
\mathcal{H}
\) will be the Hilbert space *L*^{2}(ℝ), and a suitable “resolution subspace” *V*_{0} will be chosen and assumed invariant under translation by the group of integers ℤ. In addition, it will be required that *V*_{0} be invariant under some definite scaling operator, for example under “stretching” *f* → *f*(*x*/2). Thirdly, the traditional multiresolution (MRA) approach to *L*^{2}(ℝ)-wavelets demands that the chosen subspace *V*_{0} be singly generated, i.e., generated by a single function ϕ, the father function, i.e., the normalized *L*^{2}-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 *V*_{0} are rather stringent.

## Keywords

Hilbert Space Wavelet Packet Wavelet Filter Daubechies Wavelet Dyadic Wavelet## Preview

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