# 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](src/fourier-analysis-on-finite-abelian-groups.md)
- [HA-LCA-02: Characters and the dual group](src/characters-and-the-dual-group.md)
- [HA-LCA-03: The dual group as the spectrum of \(L^1(G)\)](src/the-dual-group-as-the-gelfand-spectrum-of-l1.md)
- [HA-LCA-04: Functions of positive type](src/functions-of-positive-type.md)
- [HA-LCA-05: Raikov's theorem and the Gelfand–Raikov theorem](src/raikovs-theorem-and-the-gelfand-raikov-theorem.md)
- [HA-LCA-06: Bochner's theorem through bounded positive functionals](src/bochners-theorem.md)
- [HA-LCA-07: The Fourier inversion theorem and the dual Haar measure](src/the-fourier-inversion-theorem-and-the-dual-haar-measure.md)
- [HA-LCA-08: The Plancherel theorem](src/the-plancherel-theorem.md)
- [HA-LCA-09: The Pontryagin duality theorem](src/the-pontryagin-duality-theorem.md)
- [HA-LCA-10: Subgroups, quotients and annihilators](src/subgroups-quotients-and-annihilators.md)
- [HA-LCA-11: The Poisson summation formula](src/the-poisson-summation-formula.md)
- [HA-LCA-12: The structure of locally compact abelian groups](src/the-structure-of-locally-compact-abelian-groups.md)
- [HA-LCA-13: Unitary representations of abelian groups: the spectral theorem](src/unitary-representations-of-abelian-groups-the-spectral-theorem.md)
- [HA-LCA-14: Closed ideals of \(L^1(G)\) and Wiener's theorem](src/closed-ideals-of-l1-g-and-wieners-theorem.md)
- [HA-LCA-15: Tauberian theorems and spectral synthesis](src/tauberian-theorems-and-spectral-synthesis.md)
- [HA-LCA-16: Almost periodic functions and the Bohr compactification](src/almost-periodic-functions-and-the-bohr-compactification.md)
- [HA-LCA-17: Totally disconnected groups, the p-adic numbers and the adèles](src/totally-disconnected-groups-the-p-adic-numbers-and-the-adeles.md)

## 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](prerequisites/src/finite-radon-representation.md)
- [Spectral radius and characters of a Banach algebra](prerequisites/src/banach-spectrum.md)
- [Finite Radon products and convolution of measures](prerequisites/src/finite-radon-products.md)
- [Uniform approximation on locally compact spaces](prerequisites/src/uniform-approximation.md)
- [Integration and \(L^1\) without a countability assumption](prerequisites/src/integration-and-l1.md)
- [Haar measure on arbitrary locally compact abelian groups](prerequisites/src/haar-measure.md)
- [Hilbert spaces and unitary representations](prerequisites/src/hilbert-spaces-and-unitary-representations.md)
- [General Haar integration and the modular \(L^1\) algebra](prerequisites/src/nonabelian-haar-integration.md)
- [Dual balls and compact convexity](prerequisites/src/dual-balls-and-compact-convexity.md)
- [Real-variable calculations for the Fourier examples](prerequisites/src/real-variable-calculations.md)

## Sources and licences

Each reading identifies its exact free sources and its licence. Fremlin adaptations retain the Design Science License and the original source packages; other components retain the notices printed in them. See [component licences](LICENCES.md). The editable Markdown accompanies every rendered reading.