Fourier Series and Orthogonal Functions (Dover Books on Mathematics)
The first three chapters of the book address linear spaces, orthogonal functions, and the Fourier series. Chapter 4 introduces Legendre polynomials and Bessel functions, and Chapter 5 takes up heat and temperature. The concluding Chapter 6 explores waves and vibrations and harmonic analysis. Several topics not usually found in undergraduate texts are included, among them summability theory, generalized functions, and spherical harmonics.
Throughout the text are 570 exercises devised to encourage students to review what has been read and to apply the theory to specific problems. Those preparing for further study in functional analysis, abstract harmonic analysis, and quantum mechanics will find this book especially valuable for the rigorous preparation it provides. Professional engineers, physicists, and mathematicians seeking to extend their mathematical horizons will find it an invaluable reference as well.
Why Read This Book
You should read this book if you want a focused, mathematically clear introduction to Fourier series and orthogonal function expansions that connects theory to concrete boundary-value problems. It gives you the foundational tools used throughout spectral analysis, filter theory, and many signal-processing methods in a compact, example-driven presentation.
Who Will Benefit
Advanced undergraduates, graduate students, and practicing engineers who need a rigorous but accessible grounding in Fourier expansions, special orthogonal functions, and their use in solving PDEs and boundary-value problems.
Level: Intermediate — Prerequisites: Single-variable and multivariable calculus (including partial derivatives and multiple integrals), basic ordinary differential equations, and elementary linear algebra.
Key Takeaways
- Compute and manipulate Fourier series and understand modes of convergence (pointwise, uniform, mean-square).
- Use orthogonal function expansions to represent functions on intervals and solve projection/least-squares problems.
- Apply Sturm-Liouville theory to derive orthogonal sets and expansion coefficients for boundary-value problems.
- Work with special functions (Legendre polynomials, Bessel functions), understand their orthogonality and recurrence relations.
- Solve standard PDE boundary-value problems (heat, wave, Laplace) by Fourier/orthogonal-function methods.
Topics Covered
- 1. Linear Spaces and Inner Products
- 2. Orthogonal Functions and Series Expansions
- 3. Fourier Series: Computation and Convergence
- 4. Sturm-Liouville Problems and Eigenfunction Expansions
- 5. Legendre Polynomials and Spherical Problems
- 6. Bessel Functions and Cylindrical Problems
- 7. The Fourier Method for Boundary-Value Problems
- 8. Applications to Heat, Wave, and Laplace Equations
- Appendices: Useful Formulas and Tables
How It Compares
More applied and example-focused than advanced, measure-theoretic texts (e.g., Stein & Shakarchi); comparable in scope and audience to Brown & Churchill's Fourier Series and Boundary Value Problems but places slightly more emphasis on orthogonal functions and special functions.
