---
tag: 1GJ6SRT6B7
kind: definition
name: Harmonic analysis on groups
msc: [43A65, 43A25, 22D10]
defines: 5E6KB3V8R0
uses: [AHV05ZBCYW, VY0HXD3GC4, 74BPYJNZTM, ZR09S9R3AW, VHN5VW5Q92]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_analysis
---

## Definition

*Harmonic analysis on groups* is the part of mathematics that studies a [[locally compact group|AHV05ZBCYW]] $G$, with a [[Haar measure|VY0HXD3GC4]], through the following questions:

1. how functions in spaces such as $L^1(G)$ and $L^2(G)$ decompose into components on which $G$ acts irreducibly;
2. how the [[unitary representations|74BPYJNZTM]] of $G$, in particular the regular representation on $L^2(G)$, decompose into irreducible ones;
3. how translation, convolution and invariant differential operators act on these components.

## Remarks

Example: for a [[locally compact abelian group|ZR09S9R3AW]] $G$, the irreducible unitary representations are the continuous characters.
The decomposition is then given by the [[Fourier transform|VHN5VW5Q92]] from functions on $G$ to functions on the dual group.
For $G = \mathbb{T}$ this is the theory of Fourier series.
For compact groups the decomposition is a direct sum over finite-dimensional irreducible representations, by the Peter–Weyl theorem.
Non-example: the study of the zeros of a single polynomial on $\R$ uses no group action, and it is not harmonic analysis on groups.
The terms abstract harmonic analysis and, for non-abelian $G$, non-commutative harmonic analysis are also used.