Basic Orthogonal Wavelet Theory In Chapter 2 we saw how multiresolution analysis (MRA) works for the Haar system. A signal was decomposed into many components on different resolution levels. These components are mutually orthogonal. Despite their attractiveness, the Haar scalets and wavelets are not continuous functions. The discontinuities can create problems when applied to physical modeling. In this chapter we will construct many other orthogonal wavelets that are continuous and may even be smooth functions. . | Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright 2003 John Wiley Sons Inc. ISBN 0-471-41901-X CHAPTER THREE Basic Orthogonal Wavelet Theory In Chapter 2 we saw how multiresolution analysis MRA works for the Haar system. A signal was decomposed into many components on different resolution levels. These components are mutually orthogonal. Despite their attractiveness the Haar scalets and wavelets are not continuous functions. The discontinuities can create problems when applied to physical modeling. In this chapter we will construct many other orthogonal wavelets that are continuous and may even be smooth functions. Yet they preserve the same MRA and orthogonality as the Haar wavelets do. The wavelet basis consists of scalets Pm n t 2m 2p 2mt - n m n e Z and wavelets m n t 2m 2 2mt - n m n e Z. MULTIRESOLUTION ANALYSIS The study of orthogonal wavelets begins with the MRA. In this section we will show how an orthonormal basis of wavelets can be constructed starting from a such multiresolution analysis. Assume that a scalet p is r times differentiable with rapid decay p k t - Cpk 1 t -p k 0 1 2 . r p e Z t e R Cpk - constants. Thus we have defined a set Sr which will be used in the text p e Sr p p k t exist with rapid decay as in . 30 MULTIRESOLUTION ANALYSIS 31 A multiresolution analysis of L2 R is defined as a nested sequence of closed subspaces Vj jeZ of L2 R with the following properties 1 1 C V-i C Vo C C L2 R . 2 f e Vm f 2- e Vm i. 3 f t e Vo f t n e Vo for all n e Z. 4 Am Vm o closure Um Vm L2 R . 5 There exists p t e Vo such that set p t - n forms a Riesz basis of Vo. A Riesz basis of a separable Hilbert space H is a basis fn that is close to being orthogonal. That is there exists a bounded invertible operator which maps fn onto an orthonormal basis. Let us explain these mathematical properties intuitively In property 1 we form a nested sequence of closed subspaces. This sequence represents a causality relationship such
