Wavelets and Spaces of Functions and Distributions

In Chapter 3 we built wavelet expansions on a space of homogeneous type, which was one of main goals in this book. These wavelets are not orthonor-mal bases, but wavelet frames ψλ,λ ε Λ. These wavelets are (i) localized, (ii) smooth and (iii) oscillating. These oscillations are described by the fundamental cancellation property (iv) $\int\limits_X {\upsilon \lambda } \left( x \right)d\mu \left( x \right) = 0.$ This being said, the expansion of a function $\int \in L^2 \left( {X,d\mu } \right)$ into a wavelet series is given by

$$f\left( x \right) = \sum\limits_{\lambda \varepsilon \Lambda } {a\left( \lambda \right)} \mathop \upsilon \limits^\sim \lambda \left( x \right)$$

(4.1)

where

$$\left\| f \right\|_2 \approx \left\{ {\sum\limits_{\lambda \varepsilon \Lambda } {\left| {a\left( \lambda \right)} \right|^2 } } \right\}^{{1 \over 2}} $$

(4.2)

and where the coefficients a(λ) are given by

$$a\left( \lambda \right) = \int {f\left( x \right)} \upsilon \lambda \left( x \right)d\mu \left( x \right)$$

(4.3)

The key point is the following. The dual wavelets ψ̃λ are sharing with ψλ the same localization, smoothness and vanishing integral properties. This implies that most functional spaces will be characterized by simple size properties of the wavelet coefficients in (4.3). These wavelet expansions and characterization of functional spaces will constitute the heart of this book, which will be given in this chapter.