Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Summary of methods and applications
- 3 General methods for approximation and interpolation
- 4 Radial basis function approximation on infinite grids
- 5 Radial basis functions on scattered data
- 6 Radial basis functions with compact support
- 7 Implementations
- 8 Least squares methods
- 9 Wavelet methods with radial basis functions
- 10 Further results and open problems
- Appendix: some essentials on Fourier transforms
- Commentary on the Bibliography
- Bibliography
- Index
9 - Wavelet methods with radial basis functions
Published online by Cambridge University Press: 14 August 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Summary of methods and applications
- 3 General methods for approximation and interpolation
- 4 Radial basis function approximation on infinite grids
- 5 Radial basis functions on scattered data
- 6 Radial basis functions with compact support
- 7 Implementations
- 8 Least squares methods
- 9 Wavelet methods with radial basis functions
- 10 Further results and open problems
- Appendix: some essentials on Fourier transforms
- Commentary on the Bibliography
- Bibliography
- Index
Summary
Introduction to wavelets and prewavelets
Already in the previous chapter we have discussed in what cases L2-approximants or other smoothing methods such as quasi-interpolation or smoothing splines with radial basis functions are needed and suitable for approximation in practice, in particular when data or functions f underlying the data are at the beginning not very smooth or must be smoothed further during the computation. The so-called wavelet analysis that we will introduce now is a further development in the general context of L2-methods, and indeed everything we say here will concern L2-functions, convergence in the L2-norm etc. only. Many important books have been written on wavelets before, and since this is not at all a book on wavelets, we will be fairly short here. The reader who is interested in the specific theory of wavelets is directed to one of the excellent works on wavelets mentioned in the bibliography, for instance the books by Chui, Daubechies, Meyer and others. Here, our modest goal is to describe what wavelets may be considered as in the context of radial basis functions. The radial basis functions turn out to be useful additions to the theory of wavelets because of the versatility of the available radial basis functions.
Given a square-integrable function f on ℝ, say, the aim of wavelet analysis is to decompose it simultaneously into its time and its frequency components.
- Type
- Chapter
- Information
- Radial Basis FunctionsTheory and Implementations, pp. 209 - 230Publisher: Cambridge University PressPrint publication year: 2003