A First Course in Wavelets with Fourier Analysis, 2nd Edition

0 Inner Product Spaces.

0.1 Motivation.

0.2 Definition of Inner Product.

0.3 The Spaces L² and l².

0.4 Schwarz and Triangle Inequalities.

0.5 Orthogonality.

0.6 Linear Operators and Their Adjoints.

0.7 Least Squares and Linear Predictive Coding.

Exercises.

1 Fourier Series.

1.1 Introduction.

1.2 Computation of Fourier Series.

1.3 Convergence Theorems for Fourier Series.

Exercises.

2 The Fourier Transform.

2.1 Informal Development of the Fourier Transform.

2.2 Properties of the Fourier Transform.

2.3 Linear Filters.

2.4 The Sampling Theorem.

2.5 The Uncertainty Principle.

Exercises.

3 Discrete Fourier Analysis.

3.1 The Discrete Fourier Transform.

3.2 Discrete Signals.

3.3 Discrete Signals & Matlab.

Exercises.

4 Haar Wavelet Analysis.

4.1 Why Wavelets?

4.2 Haar Wavelets.

4.3 Haar Decomposition and Reconstruction Algorithms.

4.4 Summary.

Exercises.

5 Multiresolution Analysis.

5.1 The Multiresolution Framework.

5.2 Implementing Decomposition and Reconstruction.

5.3 Fourier Transform Criteria.

Exercises.

6 The Daubechies Wavelets.

6.1 Daubechies’ Construction.

6.2 Classification, Moments, and Smoothness.

6.3 Computational Issues.

6.4 The Scaling Function at Dyadic Points.

Exercises.

7 Other Wavelet Topics.

7.1 Computational Complexity.

7.2 Wavelets in Higher Dimensions.

7.3 Relating Decomposition and Reconstruction.

7.4 Wavelet Transform.

Appendix A: Technical Matters.

Appendix B: Solutions to Selected Exercises.

Appendix C: MATLAB® Routines.

Bibliography.

Index.