Wavelet Analysis: Basic Concepts and Applications

✍ Scribed by Sabrine Arfaoui, Anouar Ben Mabrouk, Carlo Cattani

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Tongue: English
• Leaves: 255
• Edition: 1
• Category: Library

✦ Synopsis

Wavelet Analysis: Basic Concepts and Applications provides a basic and self-contained introduction to the ideas underpinning wavelet theory and its diverse applications. This book is suitable for master’s or PhD students, senior researchers, or scientists working in industrial settings, where wavelets are used to model real-world phenomena and data needs (such as finance, medicine, engineering, transport, images, signals, etc.).

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

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
List of Figures
Preface
Chapter 1: Introduction
Chapter 2: Wavelets on Euclidean Spaces
2.1. INTRODUCTION
2.2. WAVELETS ON R
2.2.1. Continuous wavelet transform
2.2.2. Discrete wavelet transform
2.3. MULTI-RESOLUTION ANALYSIS
2.4. WAVELET ALGORITHMS
2.5. WAVELET BASIS
2.6. MULTIDIMENSIONAL REAL WAVELETS
2.7. EXAMPLES OF WAVELET FUNCTIONS AND MRA
2.7.1. Haar wavelet
2.7.2. Faber–Schauder wavelet
2.7.3. Daubechies wavelets
2.7.4. Symlet wavelets
2.7.5. Spline wavelets
2.7.6. Anisotropic wavelets
2.7.7. Cauchy wavelets
2.8. EXERCISES
Chapter 3: Wavelets extended
3.1. AFFINE GROUP WAVELETS
3.2. MULTIRESOLUTION ANALYSIS ON THE INTERVAL
3.2.1. Monasse–Perrier construction
3.2.2. Bertoluzza–Falletta construction
3.2.3. Daubechies wavelets versus Bertoluzza–Faletta
3.3. WAVELETS ON THE SPHERE
3.3.1. Introduction
3.3.2. Existence of scaling functions
3.3.3. Multiresolution analysis on the sphere
3.3.4. Existence of the mother wavelet
3.4. EXERCISES
Chapter 4: Clifford wavelets
4.1. INTRODUCTION
4.2. DIFFERENT CONSTRUCTIONS OF CLIFFORD ALGEBRAS
4.2.1. Clifford original construction
4.2.2. Quadratic form-based construction
4.2.3. A standard construction
4.3. GRADUATION IN CLIFFORD ALGEBRAS
4.4. SOME USEFUL OPERATIONS ON CLIFFORD ALGEBRAS
4.4.1. Products in Clifford algebras
4.4.2. Involutions on a Clifford algebra
4.5. CLIFFORD FUNCTIONAL ANALYSIS
4.6. EXISTENCE OF MONOGENIC EXTENSIONS
4.7. CLIFFORD-FOURIER TRANSFORM
4.8. CLIFFORD WAVELET ANALYSIS
4.8.1. Spin-group based Clifford wavelets
4.8.2. Monogenic polynomial-based Clifford wavelets
4.9. SOME EXPERIMENTATIONS
4.10. EXERCISES
Chapter 5: Quantum wavelets
5.1. INTRODUCTION
5.2. BESSEL FUNCTIONS
5.3. BESSEL WAVELETS
5.4. FRACTIONAL BESSEL WAVELETS
5.5. QUANTUM THEORY TOOLKIT
5.6. SOME QUANTUM SPECIAL FUNCTIONS
5.7. QUANTUM WAVELETS
5.8. EXERCISES
Chapter 6: Wavelets in statistics
6.1. INTRODUCTION
6.2. WAVELET ANALYSIS OF TIME SERIES
6.2.1. Wavelet time series decomposition
6.2.2. The wavelet decomposition sample
6.3. WAVELET VARIANCE AND COVARIANCE
6.4. WAVELET DECIMATED AND STATIONARY TRANSFORMS
6.4.1. Decimated wavelet transform
6.4.2. Wavelet stationary transform
6.5. WAVELET DENSITY ESTIMATION
6.5.1. Orthogonal series for density estimation
6.5.2. δ-series estimators of density
6.5.3. Linear estimators
6.5.4. Donoho estimator
6.5.5. Hall-Patil estimator
6.5.6. Positive density estimators
6.6. WAVELET THRESHOLDING
6.6.1. Linear case
6.6.2. General case
6.6.3. Local thresholding
6.6.4. Global thresholding
6.6.5. Block thresholding
6.6.6. Sequences thresholding
6.7. APPLICATION TO WAVELET DENSITY ESTIMATIONS
6.7.1. Gaussian law estimation
6.7.2. Claw density wavelet estimators
6.8. EXERCISES
Chapter 7: Wavelets for partial differential equations
7.1. INTRODUCTION
7.2. WAVELET COLLOCATION METHOD
7.3. WAVELET GALERKIN APPROACH
7.4. REDUCTION OF THE CONNECTION COEFFICIENTS NUMBER
7.5. TWO MAIN APPLICATIONS IN SOLVING PDEs
7.5.1. The Dirichlet Problem
7.5.2. The Neumann Problem
7.6. APPENDIX
7.7. EXERCISES
Chapter 8: Wavelets for fractal and multifractal functions
8.1. INTRODUCTION
8.2. HAUSDORFF MEASURE AND DIMENSION
8.3. WAVELETS FOR THE REGULARITY OF FUNCTIONS
8.4. THE MULTIFRACTAL FORMALISM
8.4.1. Frisch and Parisi multifractal formalism conjecture
8.4.2. Arneodo et al wavelet-based multifractal formalism
8.5. SELF-SIMILAR-TYPE FUNCTIONS
8.6. APPLICATION TO FINANCIAL INDEX MODELING
8.7. APPENDIX
8.8. EXERCISES
Bibliography
Index