Massopust Fractal Functions, Fractal Surfaces, and Wavelets

1. Auflage 2014
ISBN: 978-0-08-092458-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 383 Seiten, Web PDF

Fractal Functions, Fractal Surfaces, and Wavelets is the first systematic exposition of the theory of fractal surfaces, a natural outgrowth of fractal sets and fractal functions. It is also the first treatment to bring these general considerations to bear on the burgeoning field of wavelets. The text is based on Massopusts work on and contributions to the theory of fractal functions, and the author uses a number of tools--including analysis, topology, algebra, and probability theory--to introduce readers to this new subject. Though much of the material presented in this book is relatively current (developed in the past decade by the author and his colleagues) and fairly specialized, an informative background is provided for those - First systematic treatment of fractal surfaces - Links fractals and wavelets - Provides background for those entering the field - Contains color insert

Peter R. Massopust is a Privatdozent in the Center of Mathematics at the Technical University of Munich, Germany. He received his Ph.D. in Mathematics from the Georgia Institute of Technology in Atlanta, Georgia, USA, and his habilitation from the Technical University of Munich. He worked at several universities in the United States, at the Sandia National Laboratories in Albuquerque (USA), and as a senior research scientist in industry before returning to the academic environment. He has written more than sixty peer-reviewed articles in the mathematical areas of Fourier Analysis, Approximation Theory, Fractals, Splines, and Harmonic Analysis and more than 20 technical reports while working in the non-academic environment. He has authored or coauthored two textbooks and two monographs, and coedited two Contemporary Mathematics Volumes and several Special Issues for peer-reviewed journals. He is on the editorial board of several mathematics journals and has given more than one hundred invited presentations at national and international conferences, workshops, and seminars.

Autoren/Hrsg.

Massopust, Peter R.

Inhaltsverzeichnis

1;Front Cover;1
2;Fractal Functions, Fractal Surfaces, and Wavelets;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Part I:
Foundations;14
6.1;Chapter 1.
Mathematical Preliminaries;16
6.1.1;1.1 Analysis and Topology;16
6.1.2;1.2 Probability Theory;31
6.1.3;1.3 Algebra;40
6.2;Chapter 2.
Construction of Fractal Sets;54
6.2.1;2.1 Classical Fractal Sets;55
6.2.2;2.2 Iterated Function Systems;60
6.2.3;2.3 Recurrent Sets;80
6.2.4;2.4 Graph Directed Fractal Constructions;97
6.3;Chapter 3.
Dimension Theory;100
6.3.1;3.1 Topological Dimensions;100
6.3.2;3.2 Metric Dimensions;105
6.3.3;3.3 Probabilistic Dimensions;112
6.3.4;3.4 Dimension Results for Self-Affine Fractals;115
6.3.5;3.5 The Box Dimension of Projections;124
6.4;Chapter 4. Dynamical Systems and Dimension;130
7;Part II: Fractal Functions and Fractal Surfaces;146
7.1;Chapter 5. Fractal Function Construction;148
7.1.1;5.1 The Read-Bajraktarevic Operator;148
7.1.2;5.2 Recurrent Sets as Fractal Functions;159
7.1.3;5.3 Iterative Interpolation Functions;162
7.1.4;5.4 Recurrent Fractal Functions;173
7.1.5;5.5 Hidden Variable Fractal Functions;175
7.1.6;5.6 Properties of Fractal Functions;181
7.1.7;5.7 Peano Curves;200
7.1.8;5.8 Fractal Functions of Class Ck;208
7.2;Chapter 6. Dimension of Fractal Functions;218
7.2.1;6.1 Dimension Calculations;218
7.2.2;6.2 Function Spaces and Dimension;231
7.3;Chapter 7.
Fractal Functions and Wavelets;248
7.3.1;7.1 Basic Wavelet Theory;249
7.3.2;7.2 Fractal Function Wavelets;271
7.3.3;7.3 Orthogonal Fractal Function Wavelets;290
7.3.4;7.4 N-refinable Scaling Functions;313
7.4;Chapter 8. Fractal Surfaces;318
7.4.1;8.1 Tensor Product Fractal Surfaces;319
7.4.2;8.2 Affine Fractal Surfaces in Rn+m;321
7.4.3;8.3 Properties of Fractal Surfaces;331
7.4.4;8.4 Fractal Surfaces of Class Ck;352
7.5;Chapter 9.
Fractal Wavelets in Rn;360
7.5.1;9.1 Brief Review of Coxeter Groups;360
7.5.2;9.2 Fractal Functions on Foldable Figures;364
7.5.3;9.3 Interpolation on Foldable Figures;366
7.5.4;9.4 Dilation and W Invariant Spaces;368
7.5.5;9.5 Multiresolution Analyses;370
8;List of Symbols;374
9;Bibliography;378
10;Index;392