Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. This book provides students and practitioners with a practical and mathematically solid introduction to its principles. Principles of Fourier Analysis provides a comprehensive overview of the mathematical theory of Fourier analysis.
Preface; Sample Courses; I Preliminaries: 1. The Starting Point; 2. Basic Terminology, Notation and Conventions; 3. Basic Analysis I: Continuity and Smoothness; 4. Basic Analysis II: Integration and Infinite Series; 5. Symmetry and Periodicity; 6. Elementary Complex Analysis; 7. Functions of Several Variables; II Fourier Series: 8. Heuristic Derivation of the Fourier Series Formulas; 9. The Trigonometric Fourier Series; 10. Fourier Series over Finite Intervals (Sine and Cosine Series); 11. Inner Products, Norms and Orthogonality; 12. The Complex Exponential Fourier Series; 13. Convergence and Fourier's Conjecture; 14. Convergence and Fourier's Conjecture: The Proofs; 15. Derivatives and Integrals of Fourier Series; 16 Applications; III Classical Fourier Transforms: 17. Heuristic Derivation of the Classical Fourier Transform; 18. Integrals on Infinite Intervals; 19. The Fourier Integral Transforms; 20. Classical Fourier Transforms and Classically Transformable Functions; 21. Some Elementary Identities: Translation, Scaling and Conjugation; 22. Differentiation and Fourier Transforms; 23. Gaussians and Gaussian-Like Functions; 24. Convolution and Transforms of Products; 25. Correlation, Square-Integrable Functions and the Fundamental Identity; 26. Generalizing the Classical Theory: A Naive Approach; 27. Fourier Analysis in the Analysis of Systems; 28. Multi-Dimensional Fourier Transforms; 29. Identity Sequences; 30. Gaussians as Test Functions and Proofs of Important Theorems; IV Generalized Functions and Fourier Transforms: 3. A Starting Point for the Generalized Theory; 32. Gaussian Test Functions; 33. Generalized Functions; 34. Sequences and Series of Generalized Functions; 35. Basic Transforms of Generalized Fourier Analysis; 36. Generalized Products, Convolutions and Definite Integrals; 37. Periodic Functions and Regular Arrays; 38. Pole Functions and General Solutions to Simple Equations; V The Discrete Theory: 39. Periodic, Regular Arrays; 40. Sampling, Discrete Fourier Transforms and FFTs; Tables, References and Answers: Table A.1:Fourier Transforms of Some Common Functions; Table A.2: Identities for the Fourier Transforms; References; Answers to Selected Exercises
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