Fractal Functions, Fractal Surfaces, and Wavelets

Fractal Functions, Fractal Surfaces, and Wavelets

2nd Edition - August 9, 2016

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  • Author: Peter Massopust
  • Hardcover ISBN: 9780128044087
  • eBook ISBN: 9780128044704

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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.

Key Features

  • 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

Table of Contents

    • 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 si9_e
        • 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 si8_e-invariant function spaces
        • 4 Multiresolution analyses
        • 5 Wavelet sets and fractal surfaces
    • Bibliography
    • Index

Product details

  • No. of pages: 426
  • Language: English
  • Copyright: © Academic Press 2016
  • Published: August 9, 2016
  • Imprint: Academic Press
  • Hardcover ISBN: 9780128044087
  • eBook ISBN: 9780128044704

About the Author

Peter Massopust

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.

Affiliations and Expertise

Centre of Mathematics, Technical University of Munich, Germany

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  • Paulo A. Mon Mar 12 2018

    The title is clear and direct about the reader will find in the text. As a math book, with a wide and broad scope on the fractal and multifractal topics.

    In my opinion, this book is for anyone that want to deep reading about fractal in a exact mathmatics language with precise statement of theorems and mathmatics definitions that is quite hard to find in the same book. The text was written in very affordable math style.