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.
1. Cover image - p. iv 2. Title page - p. iv 3. Table of Contents - p. iv 4. Copyright - p. iv 5. Dedication - p. v 6. About the author - p. ix 7. Preface to first edition - p. xi 8. Preface to second edition - p. xv 9. List of symbols - p. xvii 10. Part I: Foundations - p. xix 11. 1: Mathematical preliminaries - p. xix 12. Abstract - p. 3 13. 1 Analysis and topology - p. 3 14. 2 Measures and probability theory - p. 15 15. 3 Algebra - p. 23 16. 4 Function spaces - p. 40 17. 2: Construction of fractal sets - p. 52 18. Abstract - p. 53 19. 1 Classical fractal sets - p. 54 20. 2 Iterated function systems - p. 59 21. 3 Local iterated function systems - p. 76 22. 4 Recurrent sets - p. 79 23. 5 Graph-directed fractal constructions - p. 94 24. 6 Transformations between fractal sets - p. 96 25. 3: Dimension theory - p. 100 26. Abstract - p. 101 27. 1 Topological dimensions - p. 101 28. 2 Metric dimensions - p. 104 29. 3 Probabilistic dimensions - p. 111 30. 4 Dimension results for self-affine fractal sets - p. 114 31. 5 The box dimension of projections - p. 122 32. 4: Dynamical systems and dimension - p. 127 33. Part II: Fractal Functions and Fractal Surfaces - p. 141 34. 5: Construction of fractal functions - p. 141 35. Abstract - p. 145 36. 1 The Read-Bajraktarević operator - p. 145 37. 2 Local fractal functions - p. 157 38. 3 Fractal bases for fractal functions - p. 163 39. 4 Recurrent sets as fractal functions - p. 166 40. 5 Iterative interpolation functions - p. 169 41. 6 Recurrent fractal functions - p. 178 42. 7 Hidden-variable fractal functions - p. 180 43. 8 Properties of fractal functions - p. 186 44. 9 Peano curves - p. 197 45. 10 Fractal functions of class Ck - p. 204 46. 11 Biaffine fractal functions - p. 210 47. 12 Local fractal functions and smoothness spaces - p. 216 48. 6: Fractels and self-referential functions - p. 230 49. 7: Dimension of fractal functions - p. 238 50. 8: Fractal functions and wavelets - p. 259 51. Abstract - p. 261 52. 1 Basic wavelet theory - p. 262 53. 2 Fractal function wavelets - p. 284 54. 3 Orthogonal fractal function wavelets - p. 302 55. 4 Wavelets are piecewise fractal functions - p. 324 56. 9: Fractal surfaces - p. 327 57. Abstract - p. 329 58. 1 Tensor product fractal surfaces - p. 329 59. 2 Affine fractal surfaces in - p. 332 60. 3 Properties of fractal surfaces - p. 343 61. 4 Fractal surfaces of class Ck - p. 366 62. 10: Fractal surfaces and wavelets in ℝn - p. 369 63. Bibliography - p. 387 64. Index - p. 397
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.
1. Cover image - p. iv 2. Title page - p. iv 3. Table of Contents - p. iv 4. Copyright - p. iv 5. Dedication - p. v 6. About the author - p. ix 7. Preface to first edition - p. xi 8. Preface to second edition - p. xv 9. List of symbols - p. xvii 10. Part I: Foundations - p. xix 11. 1: Mathematical preliminaries - p. xix 12. Abstract - p. 3 13. 1 Analysis and topology - p. 3 14. 2 Measures and probability theory - p. 15 15. 3 Algebra - p. 23 16. 4 Function spaces - p. 40 17. 2: Construction of fractal sets - p. 52 18. Abstract - p. 53 19. 1 Classical fractal sets - p. 54 20. 2 Iterated function systems - p. 59 21. 3 Local iterated function systems - p. 76 22. 4 Recurrent sets - p. 79 23. 5 Graph-directed fractal constructions - p. 94 24. 6 Transformations between fractal sets - p. 96 25. 3: Dimension theory - p. 100 26. Abstract - p. 101 27. 1 Topological dimensions - p. 101 28. 2 Metric dimensions - p. 104 29. 3 Probabilistic dimensions - p. 111 30. 4 Dimension results for self-affine fractal sets - p. 114 31. 5 The box dimension of projections - p. 122 32. 4: Dynamical systems and dimension - p. 127 33. Part II: Fractal Functions and Fractal Surfaces - p. 141 34. 5: Construction of fractal functions - p. 141 35. Abstract - p. 145 36. 1 The Read-Bajraktarević operator - p. 145 37. 2 Local fractal functions - p. 157 38. 3 Fractal bases for fractal functions - p. 163 39. 4 Recurrent sets as fractal functions - p. 166 40. 5 Iterative interpolation functions - p. 169 41. 6 Recurrent fractal functions - p. 178 42. 7 Hidden-variable fractal functions - p. 180 43. 8 Properties of fractal functions - p. 186 44. 9 Peano curves - p. 197 45. 10 Fractal functions of class Ck - p. 204 46. 11 Biaffine fractal functions - p. 210 47. 12 Local fractal functions and smoothness spaces - p. 216 48. 6: Fractels and self-referential functions - p. 230 49. 7: Dimension of fractal functions - p. 238 50. 8: Fractal functions and wavelets - p. 259 51. Abstract - p. 261 52. 1 Basic wavelet theory - p. 262 53. 2 Fractal function wavelets - p. 284 54. 3 Orthogonal fractal function wavelets - p. 302 55. 4 Wavelets are piecewise fractal functions - p. 324 56. 9: Fractal surfaces - p. 327 57. Abstract - p. 329 58. 1 Tensor product fractal surfaces - p. 329 59. 2 Affine fractal surfaces in - p. 332 60. 3 Properties of fractal surfaces - p. 343 61. 4 Fractal surfaces of class Ck - p. 366 62. 10: Fractal surfaces and wavelets in ℝn - p. 369 63. Bibliography - p. 387 64. Index - p. 397
