Fractal Functions, Fractal Surfaces, and Wavelets

Fractal Functions, Fractal Surfaces, and Wavelets, Second Edition, is the first systematic exposition of the theory of local iterated function systems, local fractal functions and fractal surfaces, and their connections to wavelets and wavelet sets. The book is based on Massopust’s work on and contributions to the theory of fractal interpolation, and the author uses a number of tools—including analysis, topology, algebra, and probability theory—to introduce readers to this exciting subject.

Though much of the material presented in this book is relatively current (developed in the past decades by the author and his colleagues) and fairly specialized, an informative background is provided for those entering the field. With its coherent and comprehensive presentation of the theory of univariate and multivariate fractal interpolation, this book will appeal to mathematicians as well as to applied scientists in the fields of physics, engineering, biomathematics, and computer science. In this second edition, Massopust includes pertinent application examples, further discusses local IFS and new fractal interpolation or fractal data, further develops the connections to wavelets and wavelet sets, and deepens and extends the pedagogical content.

• Offers a comprehensive presentation of fractal functions and fractal surfaces
• Includes latest developments in fractal interpolation
• Connects fractal geometry with wavelet theory
• Includes pertinent application examples, further discusses local IFS and new fractal interpolation or fractal data, and further develops the connections to wavelets and wavelet sets
• Deepens and extends the pedagogical content

Mathematicians working or beginning to work in the broad field of fractal geometry; physicists and engineers researching or employing fractal models; biomathematicians and computer scientists modelling fractal phenomena

• Dedication
• About the author
• Preface to first edition
• Preface to second edition
• List of symbols
• Part I: Foundations
  • 1: Mathematical preliminaries
    • Abstract
    • 1 Analysis and topology
    • 2 Measures and probability theory
    • 3 Algebra
    • 4 Function spaces
  • 2: Construction of fractal sets
    • Abstract
    • 1 Classical fractal sets
    • 2 Iterated function systems
    • 3 Local iterated function systems
    • 4 Recurrent sets
    • 5 Graph-directed fractal constructions
    • 6 Transformations between fractal sets
  • 3: Dimension theory
    • Abstract
    • 1 Topological dimensions
    • 2 Metric dimensions
    • 3 Probabilistic dimensions
    • 4 Dimension results for self-affine fractal sets
    • 5 The box dimension of projections
  • 4: Dynamical systems and dimension
    • Abstract
    • 1 Ergodic theorems and entropy
    • 2 Lyapunov dimension
• Part II: Fractal Functions and Fractal Surfaces
  • 5: Construction of fractal functions
    • Abstract
    • 1 The Read-Bajraktarević operator
    • 2 Local fractal functions
    • 3 Fractal bases for fractal functions
    • 4 Recurrent sets as fractal functions
    • 5 Iterative interpolation functions
    • 6 Recurrent fractal functions
    • 7 Hidden-variable fractal functions
    • 8 Properties of fractal functions
    • 9 Peano curves
    • 10 Fractal functions of class Ck
    • 11 Biaffine fractal functions
    • 12 Local fractal functions and smoothness spaces
  • 6: Fractels and self-referential functions
    • Abstract
    • 1 Fractels: definition and properties
    • 2 A fractel Read-Bajraktarević operator
    • 3 Further properties of fractels
  • 7: Dimension of fractal functions
    • Abstract
    • 1 Affine fractal functions
    • 2 Recurrent fractal functions
    • 3 Hidden-variable fractal functions
    • 4 Biaffine fractal functions
  • 8: Fractal functions and wavelets
    • Abstract
    • 1 Basic wavelet theory
    • 2 Fractal function wavelets
    • 3 Orthogonal fractal function wavelets
    • 4 Wavelets are piecewise fractal functions
  • 9: Fractal surfaces
    • Abstract
    • 1 Tensor product fractal surfaces
    • 2 Affine fractal surfaces in
    • 3 Properties of fractal surfaces
    • 4 Fractal surfaces of class Ck
  • 10: Fractal surfaces and wavelets in ℝn
    • Abstract
    • 1 Fractal functions on foldable figures
    • 2 Interpolation on foldable figures
    • 3 Dilation- and -invariant function spaces
    • 4 Multiresolution analyses
    • 5 Wavelet sets and fractal surfaces
• Bibliography
• Index