This volume studies representation of functions via oscillatory components such as Fourier series and transforms.
This volume studies representation of functions via oscillatory components such as Fourier series and transforms. It connects analysis, PDEs, number theory, and signal processing.
Part I. Fourier Analysis on ℝ
Chapter 1. Fourier Transform
1.1 Definition 1.2 Basic properties 1.3 Inversion formula 1.4 Plancherel theorem 1.5 Examples
Chapter 2. Convolution
2.1 Definition 2.2 Properties 2.3 Convolution theorem 2.4 Approximate identities 2.5 Applications
Chapter 3. Distributions (Overview)
3.1 Generalized functions 3.2 Fourier transform of distributions 3.3 Examples 3.4 Applications 3.5 Connections
Part II. Fourier Series
Chapter 4. Fourier Series on the Circle
4.1 Periodic functions 4.2 Fourier coefficients 4.3 Convergence 4.4 Parseval identity 4.5 Examples
Chapter 5. Convergence Theory
5.1 Pointwise convergence 5.2 Uniform convergence 5.3 Gibbs phenomenon 5.4 Applications 5.5 Examples
Chapter 6. Summability Methods
6.1 Cesàro summation 6.2 Fejér kernels 6.3 Convergence improvement 6.4 Applications 6.5 Examples
Part III. Lp Spaces and Inequalities
Chapter 7. Lp Spaces
7.1 Definitions 7.2 Norms and completeness 7.3 Duality 7.4 Interpolation (overview) 7.5 Examples
Chapter 8. Inequalities
8.1 Hölder inequality 8.2 Minkowski inequality 8.3 Young’s inequality 8.4 Hausdorff–Young inequality 8.5 Applications
Chapter 9. Singular Integrals (Overview)
9.1 Definition 9.2 Calderón–Zygmund theory (overview) 9.3 Boundedness results 9.4 Applications 9.5 Examples
Part IV. Fourier Analysis on Groups
Chapter 10. Fourier Analysis on ℝⁿ
10.1 Multidimensional transforms 10.2 Radial functions 10.3 Applications 10.4 Examples 10.5 Extensions
Chapter 11. Fourier Analysis on Compact Groups
11.1 Characters 11.2 Peter–Weyl theorem (overview) 11.3 Representation theory links 11.4 Applications 11.5 Examples
Chapter 12. Abstract Harmonic Analysis (Overview)
12.1 Locally compact groups 12.2 Haar measure 12.3 Fourier transform on groups 12.4 Applications 12.5 Examples
Part V. Time-Frequency Analysis
Chapter 13. Short-Time Fourier Transform
13.1 Definitions 13.2 Window functions 13.3 Time-frequency localization 13.4 Applications 13.5 Examples
Chapter 14. Wavelet Transform
14.1 Basic concepts 14.2 Multiresolution analysis 14.3 Continuous and discrete wavelets 14.4 Applications 14.5 Examples
Chapter 15. Frames and Bases
15.1 Frame theory 15.2 Redundant representations 15.3 Stability 15.4 Applications 15.5 Examples
Part VI. Applications
Chapter 16. PDE Applications
16.1 Fourier methods for PDEs 16.2 Heat and wave equations 16.3 Regularity analysis 16.4 Applications 16.5 Examples
Chapter 17. Signal Processing
17.1 Filtering 17.2 Sampling theory 17.3 Compression 17.4 Applications 17.5 Examples
Chapter 18. Number Theory Connections
18.1 Exponential sums 18.2 Fourier methods in number theory 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Advanced Topics
Chapter 19. Multiplier Theory
19.1 Fourier multipliers 19.2 Boundedness criteria 19.3 Applications 19.4 Examples 19.5 Connections
Chapter 20. Oscillatory Integrals
20.1 Definitions 20.2 Stationary phase method 20.3 Applications 20.4 Examples 20.5 Connections
Chapter 21. Nonlinear Harmonic Analysis
21.1 Nonlinear transforms 21.2 Applications 21.3 Examples 21.4 Connections 21.5 Emerging ideas
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Time-frequency analysis developments 22.2 Connections to geometry 22.3 Harmonic analysis on manifolds 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Convergence questions 23.2 Boundedness of operators 23.3 High-dimensional analysis 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of harmonic analysis 24.2 Key contributors 24.3 Evolution of methods 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Fourier transform identities B. Kernel formulas C. Proof techniques checklist D. Example catalog E. Cross-reference to other MSC branches
This volume develops harmonic analysis as the study of oscillatory structure. It emphasizes decomposition, transforms, and applications across mathematics and engineering.