Fourier Series
2014 Reprint of 1962 Edition. Full facsimile of the original edition. Not reproduced with Optical Recognition Software. The present volume is an introduction to Fourier series and their use in solving boundary value problems of mathematical physics. The text treats expansions in Fourier series, general orthogonal expansions, convergence of Fourier series, operations with Fourier series, double Fourier series, Fourier integrals and transforms, Bessel functions and Fourier-Bessel series, the eigenfunction method and its use in solving boundary value problems of mathematical analysis, applications to vibrating systems and heat flow problems. Every chapter moves clearly from topic to topic and theorem to theorem, with many theorem proofs given. A total of 107 problems will be found at the ends of the chapters, including many specially added to this English-language edition, and answers are given at the end of the text. Tolstov was one of the foremost mathematicians of the former Soviet Union.
Why Read This Book
You will gain a rigorous, classical grounding in Fourier series and related integral transforms that underpins modern spectral methods used in DSP, communications, and radar. Tolstov’s clear progression from theory to boundary-value applications helps you connect convergence theorems and orthogonal expansions to practical problems like heat flow, vibrations, and modal decompositions used in signal analysis.
Who Will Benefit
Advanced undergraduates, graduate students, and practicing engineers in DSP, audio/speech, radar, or communications who need a mathematically precise foundation in Fourier methods for solving PDEs and analyzing spectra.
Level: Advanced — Prerequisites: Single- and multivariable calculus, ordinary differential equations, basic partial differential equations, linear algebra; familiarity with complex numbers and elementary real analysis is highly recommended.
Key Takeaways
- Understand the formulation and convergence criteria for Fourier series and orthogonal expansions.
- Apply eigenfunction methods and Fourier-Bessel series to solve boundary value problems in heat and wave equations.
- Derive and use Fourier integrals and transforms to move between time/space and frequency domains.
- Analyze double Fourier series and operations (differentiation, integration, termwise manipulation) with rigorous justification.
- Connect classical Fourier theory to spectral analysis concepts used in filtering, modal decomposition, and problems with cylindrical symmetry.
Topics Covered
- 1. Introduction and Historical Background
- 2. Expansions in Fourier Series
- 3. Convergence of Fourier Series
- 4. Operations with Fourier Series (termwise differentiation, integration, summation methods)
- 5. Double Fourier Series and Multiple Fourier Expansions
- 6. Fourier Integrals and the Fourier Transform
- 7. Bessel Functions and Fourier–Bessel Series
- 8. The Eigenfunction Method for Boundary Value Problems
- 9. Applications to Vibrating Systems and Heat Flow
- 10. Problems and Exercises (107 problems)
- Appendices: Useful Formulas and Tables
How It Compares
More mathematically focused than engineering DSP texts such as Oppenheim & Schafer's Signals and Systems (which emphasize discrete-time and implementation), Tolstov is closer in spirit to classical treatments like Stein & Shakarchi or Titchmarsh but with more emphasis on PDE applications and worked problems.
