Wavelets, Their Friends, and What They Can Do for You
The goal of these lecture notes is to introduce the reader to the central concepts related to wavelets and their applications as quickly as possible. By focusing on essential ideas and arguments, the authors indicate appropriate places in the literature for detailed proofs and real applications. The book is organised as follows. A preliminary chapter containing some of the required concepts and definitions is included for reference. In chapter 2, time-frequency analysis is reviewed. The authors start with Fourier series and the Fourier transform. The windowed Fourier transform, the Gabor basis and local trigonometric bases are discussed. Finally the authors gain localization at the scales with the wavelet transform. In chapter 3, the notion of multiresolution analysis (MRA) is introduced. The Haar wavelets and MRA is carefully described. This example contains most of the important ideas behind MRA. Daubechies compactly supported wavelets are described briefly and pointers to the literature are given.
In chapter 4, the authors discuss variants of the classical orthogonal wavelets and MRA. The MRA associated to biorthogonal and multiwavelets are also carefully described. The authors also explain how to construct wavelets in two dimensions by tensor products. The wavelets confined to an interval or a domain in the space are very briefly mentioned. The authors also discuss wavelet packets and some relatives and mutations of wavelets that have been constructed to tackle more specialized problems. Finally the prolate spheroidal wave functions are studied. In chapter 5, a few applications are described without attempting to be systematic or comprehensive. The choice of sample applications is dictated by the authors’ experiences in this area. A description is made in some detail of how to calculate derivatives using biorthogonal wavelets, and how one can construct wavelets with more fancy differential properties. The authors also describe how wavelets can characterize a variety of function spaces and how well adapted wavelets are able to identify very fine local regularity properties of functions. Finally the authors very briefly describe how wavelets could be used for the study of differential equations. The authors expect the reader to have been exposed to some real and complex analysis, calculus and linear algebra and some concepts of orthonormal bases, orthogonal projections and orthogonal complements on Hilbert space. The book is intended for beginning graduate students and beyond.