Introduction to Fourier Analysis and Wavelets (Brooks/Cole Series in Advanced Mathematics)
Written by a successful author and respected mathematician, this book emphasizes a concrete and computational approach to the subject of Fourier analysis and wavelet theory while maintaining a balance between theory and applications. In some cases, several different proofs are offered for a given proposition, allowing students to compare different methods.
Why Read This Book
You should read this book if you want a rigorous but accessible bridge between classical Fourier analysis and modern wavelet methods, presented with computational examples and multiple proof approaches so you can see both theory and practice. It gives you the mathematical tools to understand why transforms and wavelet bases work and how to apply them to signal analysis.
Who Will Benefit
Advanced undergraduates, graduate students, and practicing engineers who need a solid theoretical foundation in Fourier methods and wavelets to support DSP work or research.
Level: Intermediate — Prerequisites: Calculus (including series and integrals), basic linear algebra, and familiarity with elementary real analysis or signals and systems concepts.
Key Takeaways
- Understand the theory and convergence properties of Fourier series and the Fourier transform.
- Apply Parseval/Plancherel identities and distributional viewpoints to analyze energy and spectral content.
- Derive and use sampling relations and Poisson summation for spectral analysis and reconstruction.
- Explain multiresolution analysis and construct scaling functions and orthonormal wavelet bases.
- Implement and reason about discrete wavelet transforms and their underlying filter-bank structure.
- Analyze continuous wavelet transforms and understand practical trade-offs between time and frequency localization.
Topics Covered
- 1. Introduction and motivations
- 2. Fourier Series: definitions and examples
- 3. Convergence, Gibbs phenomenon, and summability methods
- 4. The Fourier Transform in L^1 and L^2
- 5. Plancherel, Parseval, and distributional viewpoints
- 6. Sampling, the Poisson summation formula, and Shannon sampling
- 7. Time-frequency ideas (short-time Fourier ideas and preliminaries)
- 8. Introduction to Wavelets and Multiresolution Analysis (MRA)
- 9. Scaling functions, filter relations, and refinement equations
- 10. Construction of orthonormal wavelet bases (Haar, Daubechies examples)
- 11. Discrete Wavelet Transform, filter banks, and algorithms
- 12. Continuous Wavelet Transform and applications
- 13. Computational examples, exercises, and further directions
How It Compares
More math-focused than Mallat's A Wavelet Tour of Signal Processing (which is application-heavy) and more accessible than Daubechies' Ten Lectures on Wavelets while still giving solid theoretical grounding.
