Wavelet Analysis: Basic Concepts and Applications
254
Wavelet Analysis: Basic Concepts and Applications
254Hardcover
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Overview
Features:
- Offers a self-contained discussion of wavelet theory
- Suitable for a wide audience of post-graduate students, researchers, practitioners, and theorists
- Provides researchers with detailed proofs
- Provides guides for readers to help them understand and practice wavelet analysis in different areas
Product Details
| ISBN-13: | 9780367562182 |
|---|---|
| Publisher: | CRC Press |
| Publication date: | 04/21/2021 |
| Pages: | 254 |
| Product dimensions: | 7.00(w) x 10.00(h) x (d) |
About the Author
Anouar Ben Mabrouk is currently working as the professor of mathematics. He is also the associate professor of Mathematics at the University of Kairouan, Tunisia, the Faculty of Sciences, University of Monastir. His main research interests are are wavelets, fractals, probability/statistics, PDEs and related fields such as financial mathematics, time series, image/signal processing, numerical and theoretical aspects of PDEs. Dr. Ben Mabrouk is currently associated with the University of Tabuk, Saudi Arabia in a technical cooperation project.
Carlo Cattani is currently the professor of Mathematical Physics and Applied Mathematics at the Engineering School (DEIM) of University of Tuscia. His scientific interests include but are not limited to wavelets, dynamical systems, fractals, fractional calculus, numerical methods, number theory, stochastic integro-differential equations, competition models, time-series analysis, nonlinear analysis, complexity of living systems, pattern analysis, computational biology, biophysics, history of science. He has (co)authored more than 150 scientific articles on international journals as well as several books.
Table of Contents
List of Figures ix
Preface xi
Chapter 1 Introduction 1
Chapter 2 Wavelets on Euclidean Spaces 5
2.1 Introduction 5
2.2 Wavelets One 6
2.2.1 Continuous wavelet transform 7
2.2.2 Discrete wavelet transform 10
2.3 Multi-Resolution Analysis 11
2.4 Wavelet Algorithms 13
2.5 Wavelet Basis 16
2.6 Multidimensional Real Wavelets 21
2.7 Examples Of Wavelet Functions and MRA 22
2.7.1 Haar wavelet 22
2.7.2 Faber-Schauder wavelet 24
2.7.3 Daubechies wavelets 25
2.7.4 Symlet wavelets 27
2.7.5 Spline wavelets 27
2.7.6 Anisotropic wavelets 29
2.7.7 Cauchy wavelets 30
2.8 Exercises 31
Chapter 3 Wavelets extended 35
3.1 Affine Group Wavelets 35
3.2 Multiresolution Analysis on the Interval 37
3.2.1 Monasse-Perrier construction 37
3.2.2 Bertoluzza-Falletta construction 37
3.2.3 Daubechies wavelets versus Bertoluzza-Faletta 39
3.3 Wavelets on the Sphere 40
3.3.1 Introduction 40
3.3.2 Existence of scaling functions 41
3.3.3 Multiresolution analysis on the sphere 43
3.3.4 Existence of the mother wavelet 44
3.4 Exercises 47
Chapter 4 Clifford wavelets 51
4.1 Introduction 51
4.2 Different Constructions of Clifford Algebras 52
4.2.1 Clifford original construction 53
4.2.2 Quadratic form-based construction 53
4.2.3 A standard construction 54
4.3 Graduation in Clifford Algebras 56
4.4 Some Useful Operations on Clifford Algebras 57
4.4.1 Products in Clifford algebras 57
4.4.2 Involutions on a Clifford algebra 58
4.5 Clifford Functional Analysis 60
4.6 Existence of Monogenic Extensions 67
4.7 Clifford-Fourier Transform 70
4.8 Clifford Wavelet Analysis 76
4.8.1 Spin-group based Clifford wavelets 76
4.8.2 Monogenic polynomial-based Clifford wavelets 82
4.9 Some Experimentations 92
4.10 Exercises 96
Chapter 5 Quantum wavelets 99
5.1 Introduction 99
5.2 Bessel Functions 99
5.3 Bessel Wavelets 105
5.4 Fractional Bessel Wavelets 107
5.5 Quantum Theory Toolkit 119
5.6 Some Quantum Special Functions 123
5.7 Quantum Wavelets 127
5.8 Exercises 134
Chapter 6 Wavelets in statistics 137
6.1 Introduction 137
6.2 Wavelet Analysis of Time Series 138
6.2.1 Wavelet time series decomposition 138
6.2.2 The wavelet decomposition sample 140
6.3 Wavelet Variance and Covariance 141
6.4 Wavelet Decimated and Stationary Transforms 144
6.4.1 Decimated wavelet transform 144
6.4.2 Wavelet stationary transform 145
6.5 Wavelet Density Estimation 145
6.5.1 Orthogonal series for density estimation 145
6.5.2 δ-series estimators of density 147
6.5.3 Linear estimators 148
6.5.4 Donoho estimator 150
6.5.5 Hall-Patil estimator 150
6.5.6 Positive density estimators 151
6.6 Wavelet Thresholding 152
6.6.1 Linear case 152
6.6.2 General case 154
6.6.3 Local thresholding 155
6.6.4 Global thresholding 155
6.6.5 Block thresholding 156
6.6.6 Sequences thresholding 156
6.7 Application to Wavelet Density Estimations 157
6.7.1 Gaussian law estimation 158
6.7.2 Claw density wavelet estimators 159
6.8 Exercises 160
Chapter 7 Wavelets for partial differential equations 163
7.1 Introduction 163
7.2 Wavelet Collocation Method 165
7.3 Wavelet Galerkin Approach 166
7.4 Reduction of the Connection Coefficients Number 171
7.5 Two Main Applications in Solving PDEs 174
7.5.1 The Dirichlet Problem 174
7.5.2 The Neumann Problem 176
7.6 Appendix 179
7.7 Exercises 180
Chapter 8 Wavelets for fractal and multifractal functions 183
8.1 Introduction 183
8.2 Hausdorff Measure and Dimension 184
8.3 Wavelets for the Regularity of Functions 186
8.4 The Multifractal Formalism 189
8.4.1 Frisch and Parisi multifractal formalism conjecture 189
8.4.2 Araeodo et al wavelet-based multifractal formalism 190
8.5 Self-Similar-Type Functions 192
8.6 Application to Financial Index Modeling 201
8.7 Appendix 205
8.8 Exercises 205
Bibliography 209
Index 237