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The idea of modeling the behaviour of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describe nonlinear phenomena better than traditional mathematical models. In many cases they have been used for solving inverse problems arising in models described by systems of differential equations and dynamical systems.

"Fractal-Based Methods in Analysis" draws together, for the first time in book form, methods and results from almost twenty years of research in this topic, including new viewpoints and results in many of the chapters. For each topic the theoretical framework is carefully explained using examples and applications.

The second chapter on basic iterated function systems theory is designed to be used as the basis for a course and includes many exercises. This chapter, along with the three background appendices on topological and metric spaces, measure theory, and basic results from set-valued analysis, make the book suitable for self-study or as a source book for a graduate course. The other chapters illustrate many extensions and applications of fractal-based methods to different areas. This book is intended for graduate students and researchers in applied mathematics, engineering and social sciences.

Herb Kunze is a professor of mathematics at the University of Guelph in Ontario. Davide La Torre is an associate professor of mathematics in the Department of Economics, Management and Quantitative Methods of the University of Milan. Franklin Mendivil is a professor of mathematics at Acadia University in Nova Scotia. Edward Vrscay is a professor in the department of Applied Mathematics at the University of Waterloo in Ontario. The major focus of their research is on fractals and the applications of fractals.

Inhaltsverzeichnis

Frontmatter

Chapter 1. What do we mean by “Fractal-Based Analysis”?

Abstract
We consider “fractal-based analysis” to be the mathematics (and applications) associated with two fundamental ideas, namely, (1) selfsimilarity and (2) contractivity:
Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

Chapter 2. Basic IFS Theory

Abstract
In this chapter, we give a short presentation of the classical theory of iterated function systems (IFSs). Our main purpose in doing this is to make this book as self-contained as possible. We also hope that this chapter might be used by a beginner to learn this theory. As such, the tone in this chapter is more expository than in some of the other chapters, and we give more attention to pedagogical motivation.
Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

Chapter 3. IFS on Spaces of Functions

Abstract
Geometric and measure-theoretic IFSs can easily be extended to IFS operators acting on functions. These operators are closely related to the IFS on measures from Sect. 2.5. Historically, one primary motivation for these operators was the desire to represent digital images by means of attractors of IFSs.
Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

Chapter 4. Iterated Function Systems, Multifunctions, and Measure-Valued Functions

Abstract
In this chapter, we expand on the construction of an IFS with probability (as seen in Sect. 2.5 in Chapter 2). We do this in many different directions, but all with the same overall scheme in mind.
Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

Chapter 5. IFS on Spaces of Measures

Abstract
In this chapter, we expand on the construction of an IFS with probability (as seen in Sect. 2.5 in Chapter 2). We do this in many different directions, but all with the same overall scheme in mind.
Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

Chapter 6. The Chaos Game

Abstract
We saw the chaos game in Chapter 2, where it was introduced first as a means of generating an image of the attractor of an IFS in R2. In this chapter, we will see several other things one can do with the chaos game. First we will modify the chaos game to obtain a way of generating approximations of the invariant function for an IFSM (see Chapter 3 for the basic properties and results about an IFS on functions). Our modification is inspired by work of Berger [21, 22, 23], who constructed a chaos game for generating the graph of a wavelet.
Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

Chapter 7. Inverse Problems and Fractal-Based Methods

Abstract
In this chapter, we consider an assortment of inverse problems for differential and integral equations, all of which can be treated within the framework of Banach’s fixed point theorem and the collage theorem. As always, the essence of the method is the approximation of elements of a complete metric space by fixed points of contractive operators on that space.
Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

Chapter 8. Further Developments and Extensions

Abstract
In this section, we formulate collage theorems to treat inverse problems for elliptic, parabolic, and hyperbolic partial differential equations. Such problems have understandably received a great deal of attention over the years in the study of distributed systems (see, for example, [142]).
Herb Kunze, Davide La Torre, Franklin Mendivil, Edward R. Vrscay

Backmatter

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