---
tag: 5E6KB3V8R0
kind: concept
name: Harmonic analysis on groups
msc: [43A65, 43A25, 22D10]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_analysis
---

## Summary

Harmonic analysis on groups studies functions on a locally compact group by writing them as superpositions of basic oscillations attached to the group.
For abelian groups the oscillations are the characters and the main tool is the Fourier transform on the dual group.
For non-abelian groups the characters are replaced by irreducible unitary representations.
The subject contains Fourier series, the Fourier transform on $\R^n$, and the representation theory of compact and Lie groups.

## Explanations

### L1 human

A musical sound can be split into pure tones; harmonic analysis does the same for functions.
Which "pure tones" are available depends on the symmetry of the space: on a circle they are the waves $e^{in\theta}$, on the line they are $e^{i\xi x}$.
On a sphere, where rotations act, the pure tones are spherical harmonics, and they come from representations of the rotation group.
Harmonic analysis on groups is the general study of this splitting for any group with a reasonable topology.