This volume extends harmonic analysis to general locally compact groups.
This volume extends harmonic analysis to general locally compact groups. It replaces classical Fourier series and transforms with representation-theoretic and measure-theoretic frameworks.
Part I. Locally Compact Groups
Chapter 1. Topological Groups
1.1 Definitions and examples 1.2 Continuity and group operations 1.3 Subgroups and quotient groups 1.4 Compact and locally compact groups 1.5 Examples
Chapter 2. Haar Measure
2.1 Existence and uniqueness 2.2 Left and right invariance 2.3 Integration on groups 2.4 Modular function 2.5 Examples
Chapter 3. Function Spaces on Groups
3.1 Continuous functions 3.2 Lp spaces on groups 3.3 Convolution algebras 3.4 Approximate identities 3.5 Examples
Part II. Representations of Groups
Chapter 4. Unitary Representations
4.1 Definitions 4.2 Hilbert space representations 4.3 Irreducible representations 4.4 Direct sums and integrals 4.5 Examples
Chapter 5. Dual Objects
5.1 Dual group for abelian groups 5.2 Characters 5.3 Pontryagin duality 5.4 Examples 5.5 Applications
Chapter 6. Non-Abelian Representation Theory
6.1 Representations of non-abelian groups 6.2 Peter–Weyl theorem 6.3 Matrix coefficients 6.4 Applications 6.5 Examples
Part III. Fourier Analysis on Groups
Chapter 7. Fourier Transform on Groups
7.1 Definition 7.2 Properties 7.3 Inversion formula 7.4 Plancherel theorem 7.5 Examples
Chapter 8. Convolution and Algebras
8.1 Convolution operators 8.2 Group algebras 8.3 Banach algebra structure 8.4 Applications 8.5 Examples
Chapter 9. Spectral Theory
9.1 Spectra of operators 9.2 Functional calculus 9.3 Applications 9.4 Examples 9.5 Connections
Part IV. Structure of Groups
Chapter 10. Compact Groups
10.1 Structure theory 10.2 Representations 10.3 Harmonic analysis 10.4 Applications 10.5 Examples
Chapter 11. Abelian Groups
11.1 Structure theorem 11.2 Fourier analysis simplifications 11.3 Duality 11.4 Applications 11.5 Examples
Chapter 12. Non-Abelian Groups
12.1 Structural challenges 12.2 Representation decomposition 12.3 Harmonic analysis 12.4 Applications 12.5 Examples
Part V. Operator Algebras
Chapter 13. Group C*-Algebras
13.1 Definitions 13.2 Representations 13.3 Properties 13.4 Applications 13.5 Examples
Chapter 14. Von Neumann Algebras
14.1 Definitions 14.2 Factors 14.3 Representations 14.4 Applications 14.5 Examples
Chapter 15. Noncommutative Integration
15.1 Operator-valued integration 15.2 Trace and states 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Applications
Chapter 16. PDE and Analysis
16.1 Harmonic analysis methods 16.2 Differential operators 16.3 Applications 16.4 Examples 16.5 Connections
Chapter 17. Number Theory
17.1 Automorphic forms (overview) 17.2 Representation theory links 17.3 Applications 17.4 Examples 17.5 Connections
Chapter 18. Physics and Engineering
18.1 Symmetry analysis 18.2 Signal processing on groups 18.3 Quantum mechanics 18.4 Applications 18.5 Examples
Part VII. Advanced Topics
Chapter 19. Induced Representations
19.1 Definition 19.2 Mackey theory (overview) 19.3 Applications 19.4 Examples 19.5 Connections
Chapter 20. Noncommutative Harmonic Analysis
20.1 General framework 20.2 Applications 20.3 Examples 20.4 Connections 20.5 Emerging ideas
Chapter 21. Harmonic Analysis on Homogeneous Spaces
21.1 Quotient spaces 21.2 Invariant measures 21.3 Representations 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Representation theory developments 22.2 Noncommutative geometry 22.3 Harmonic analysis on manifolds 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Representation classification 23.2 Spectral analysis challenges 23.3 Computational aspects 23.4 Analytical difficulties 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of abstract harmonic analysis 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Standard group examples B. Key theorems summary C. Proof techniques checklist D. Operator algebra reference E. Cross-reference to other MSC branches
This volume generalizes harmonic analysis to abstract group settings. It emphasizes representation theory, operator algebras, and invariant integration.