Fractal Functions, Fractal Surfaces, and Wavelets

This monograph gives an introduction to the theory of fractal functions and fractal surfaces with an application to wavelet theory. The study of fractal functions goes back to WeierstraB’s nowhere differentiable function and beyond. However, it wasn’t until the publication of B. Mandelbrot’s book (cf. [123]) in which the concept of a fractal set was introduced and common characteristics of these sets were identified (such as nonintegral dimension and geometric self-similarity) that the theory of functions with fractahgraphs developed into an area of its own. Seemingly different types of nowhere differentiable functions, such as those investigated by Besicovitch, Ursell, Knopp, and Kiesswetter, to only mention a few, were unified under the fractal point of view. This unification led to new mathematical methods and applications in areas that include: dimension theory, dynamical systems and chaotic dynamics, image analysis, and wavelet theory. The objective of this monograph is to provide essential results from the theory of fractal functions and surfaces for those interested in this fascinating area, to present new and exciting applications, and to indicate which interesting directions the theory can be extended. The book is essentially self-contained and covers the basic theory and different types of fractal constructions as well as some specialized and advanced topics such as dimension calculations and function space theory. The first part of the book contains background material and consists of four chapters. The first chapter introduces the relevant notation and terminology and gives a brief review of some of the basic concepts from classical analysis, abstract algebra and probability theory that are necessary for the remainder of the book. The reader who is not quite familiar with some of the material presented in this first chapter is referred to the bibliography where most of these concepts are defined and motivated. However, efforts were made to keep the mathematical requirements at a level where a graduate student