Fourier Analysis and Its Applications (Pure and Applied Undergraduate Texts)
This book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.
Why Read This Book
You will gain a clear, mathematically grounded toolkit for using Fourier series, transforms, and eigenfunction expansions to solve real engineering problems in DSP, communications, and waves. The book balances modern-analysis ideas with many physics and engineering applications so you can understand not just how to compute transforms but why the methods work and when to apply them.
Who Will Benefit
Advanced undergraduates, beginning graduate students, and practicing engineers who want a rigorous yet application-focused foundation in Fourier methods for signal processing, communications, and wave problems.
Level: Advanced — Prerequisites: Multivariable calculus, ordinary differential equations, and linear algebra; familiarity with basic real analysis or complex numbers is highly recommended.
Key Takeaways
- Apply Fourier series and integrals to analyze and solve boundary-value problems that arise in signal processing and wave physics
- Derive and use Fourier transforms, convolution theorems, and Plancherel/Parseval identities for spectral analysis of signals
- Use eigenfunction expansions (Sturm–Liouville theory) and orthogonal functions to decompose signals and solve PDEs
- Work with Bessel functions, Hankel transforms, and related special functions useful in radar, acoustics, and circular/spherical geometries
- Use Laplace transforms, generalized functions (distributions), and Green's functions to treat impulsive signals and solve ODE/PDE initial-value problems
- Connect rigorous theoretical results to practical engineering contexts (filters, spectral estimation, modal expansions) so you can choose appropriate analytical tools
Topics Covered
- 1. Introduction and basic examples
- 2. Fourier series on intervals and convergence issues
- 3. Orthogonal expansions and Sturm–Liouville theory
- 4. The Fourier transform on R^n: definitions and properties
- 5. Convolution, Plancherel theorem, and applications to LTI systems
- 6. Special functions: Bessel functions and Hankel transforms
- 7. Orthogonal polynomials and further eigenfunction systems
- 8. Laplace transforms and applications to initial-value problems
- 9. Generalized functions and distributional Fourier transforms
- 10. Green's functions for ordinary differential equations
- 11. Green's functions and fundamental solutions for partial differential equations
- 12. Selected applications to physics and engineering (waves, acoustics, heat, diffusion)
- Appendices: auxiliary results and proofs
How It Compares
More mathematically rigorous and broader in scope than Bracewell's The Fourier Transform and Its Applications, and more application-oriented and expansive in special functions and Green's functions than Stein & Shakarchi's Fourier Analysis.
