Harmonic analysis on locally compact abelian groups

Edition of 4 October 2026.

A complete course in 17 lessons, with ten prerequisite readings and 454 proved results and solved exercises. The course treats arbitrary locally compact abelian groups, including groups without countability or sigma-finiteness assumptions. Its final lessons develop spectral theory, Tauberian theorems, spectral synthesis, almost periodic functions, and p-adic and adelic analysis.

The mathematical sources are verified freely accessible works. Every used result is proved in this edition or an exact earlier reading within it. Self-checked by the writing AI.

Lessons

• HA-LCA-01: Fourier analysis on finite abelian groups
• HA-LCA-02: Characters and the dual group
• HA-LCA-03: The dual group as the spectrum of \(L^1(G)\)
• HA-LCA-04: Functions of positive type
• HA-LCA-05: Raikov's theorem and the Gelfand–Raikov theorem
• HA-LCA-06: Bochner's theorem through bounded positive functionals
• HA-LCA-07: The Fourier inversion theorem and the dual Haar measure
• HA-LCA-08: The Plancherel theorem
• HA-LCA-09: The Pontryagin duality theorem
• HA-LCA-10: Subgroups, quotients and annihilators
• HA-LCA-11: The Poisson summation formula
• HA-LCA-12: The structure of locally compact abelian groups
• HA-LCA-13: Unitary representations of abelian groups: the spectral theorem
• HA-LCA-14: Closed ideals of \(L^1(G)\) and Wiener's theorem
• HA-LCA-15: Tauberian theorems and spectral synthesis
• HA-LCA-16: Almost periodic functions and the Bohr compactification
• HA-LCA-17: Totally disconnected groups, the p-adic numbers and the adèles

Prerequisite readings

The real-variable reading follows Lesson 2 and precedes Lesson 3. The remaining readings supply the topology, measure, algebra and Hilbert-space foundations used in the lessons; exact proof links identify each dependency.

• Finite Radon representation on locally compact spaces
• Spectral radius and characters of a Banach algebra
• Finite Radon products and convolution of measures
• Uniform approximation on locally compact spaces
• Integration and \(L^1\) without a countability assumption
• Haar measure on arbitrary locally compact abelian groups
• Hilbert spaces and unitary representations
• General Haar integration and the modular \(L^1\) algebra
• Dual balls and compact convexity
• Real-variable calculations for the Fourier examples

Sources and licences

Each reading identifies its exact free sources and its licence. The readings are CC0; those that follow Fremlin's arguments cite him, and his unchanged source packages keep their own licence. See component licences. The editable Markdown accompanies every rendered reading.