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Wavelets and Spaces of Functions and Distributions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1966)

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.

Keywords

  • Hardy Space
  • Besov Space
  • Continuous Linear Operator
  • Wavelet Frame
  • Homogeneous Type

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2009 Springer-Verlag Berlin Heidelberg

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(2009). Wavelets and Spaces of Functions and Distributions. In: Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics, vol 1966. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88745-4_5

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