Abstract
This chapter presents a brief introduction to wavelet theory, developing the necessary notation and terminology needed later. It also presents the basic concepts of wavelet theory: refinable shift-invariant function spaces, scaling functions, multiresolution analyses, and reconstruction and decomposition algorithms. There are two approaches to obtain wavelet decompositions. The first uses the concept of multiresolution analysis. This multiresolution analysis yields, in general, a finite set of scaling functions that are then used to define the wavelets and the wavelet decompositions. The second method begins with a dilation equation for a given function or set of functions. The nontrivial solutions of this dilation equation then give scaling functions that generate a multiresolution analysis. The chapter presents a construction of wavelets based on fractal functions and presents a method for constructing stable shift- and dilation-invariant function spaces using the fractal functions.
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