Chapter X. Introduction to Wavelets

Abstract

This chapter introduces the relatively recent subject of wavelets, which is an outgrowth of Fourier analysis in mathematics and signal processing in engineering. Except in one case, construction of examples of wavelets tends to be difficult. Much of the chapter is devoted to construction of some of the better known examples and lists of their most important properties.

Section 1 defines wavelets and discusses three features of traditional Fourier analysis: the Uncertainty Principle, Gibbs phenomenon, and the Shannon Sampling Theorem. It ends with a brief essay on the need for wavelets in various applications.

Section 2 establishes that the Haar system is an orthonormal basis of $L^{2}(\mathbb {R})$. The Haar wavelet predates the general theory of wavelets by many decades but provides a prototype for some of the development. The section ends with some discussion of convergence of one-sided Haar expansions for function spaces besides $L^{2}(\mathbb {R})$.

Section 3 begins the general theory of wavelets, introducing the notion of multiresolution analysis to abstract the construction in Section 2 of the Haar wavelet. The ingredients of a multiresolution analysis are a scaling function, traditionally called $\varphi$, and an increasing sequence of closed subspaces $V_{j}$ of $L^{2}(\mathbb {R})$ with certain properties. The wavelet that is constructed is traditionally called $\psi$.

Section 4 introduces the Shannon wavelet, whose construction is immediate from the theory of multiresolution analyses. The new ingredient here, beyond the ideas used for the Haar wavelet, is the careful use of the generating function of the scaling function to obtain a formula for the wavelet.

Section 5 supplements the theory of Section 3 by showing how to build a multiresolution analysis out of a candidate for the scaling function.

Section 6 introduces the Meyer wavelets, each of which is smooth and has Fourier transform of a prescribed order of differentiability. The full theory of Sections 3 and 5 is used in their construction.

Section 7 introduces splines, examines one example, and sees the need for more theory. It develops one further aspect of the general theory, showing how to replace a “Riesz system” with an orthonormal set. It therefore allows one to relax the conditions needed in Section 5 for a function to be a scaling function. In addition, it uses elementary complex analysis to prove a series expansion for $\pi^{2}/\sin^{2}\pi z$ that is needed in Section 8.

Section 8 continues the discussion of the role of splines in the theory of wavelets, introducing the Battle–Lemarié wavelets. As with the Meyer wavelets each is smooth and has Fourier transform of a prescribed order of differentiability.

Section 9 develops the Daubechies wavelets. These have compact support, but except for the first one, neither they nor their Fourier transforms have known formulas in closed form. The construction begins by pinpointing necessary conditions on the generating function.

Section 10 deals with smoothness questions. It contains three results. The first gives an estimate for the decay of the Fourier transform of the Daubechies scaling function of each order. The second deduces a certain amount of differentiability of a scaling function from the estimate in the first result. The third shows in the converse direction that a Daubechies wavelet can never be of class $C^{∞}$. The section concludes with a table summarizing properties of the specific wavelets that have been constructed in Sections 2–9.

Section 11 gives a quick introduction to applications. It discusses the discrete wavelet transform and its use in storage and compression of data, it identifies some applications of wavelets in one and two dimensions, and it makes brief remarks about some of the applications.