Math 2.0

Abstract harmonic analysis Definition

Harmonic analysis on groups

Builds on Locally compact groupHaar measureUnitary representationLocally compact abelian groupFourier transform on locally compact abelian groups

Definition

Harmonic analysis on groups is the part of mathematics that studies a locally compact group GG, with a Haar measure, through the following questions:

  1. how functions in spaces such as L1(G)L^1(G) and L2(G)L^2(G) decompose into components on which GG acts irreducibly;
  2. how the unitary representations of GG, in particular the regular representation on L2(G)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 GG, the irreducible unitary representations are the continuous characters. The decomposition is then given by the Fourier transform from functions on GG to functions on the dual group. For G=𝕋G = \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\R uses no group action, and it is not harmonic analysis on groups. The terms abstract harmonic analysis and, for non-abelian GG, non-commutative harmonic analysis are also used.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Fourier transform on locally compact abelian groups, concept, no evidence levelFourier transform on …conceptHaar measure, concept, no evidence levelHaar measureconceptLocally compact abelian group, concept, no evidence levelLocally compact abeli…conceptLocally compact group, concept, no evidence levelLocally compact groupconceptUnitary representation, concept, no evidence levelUnitary representationconceptHarmonic analysis on groups, definition, no evidence levelHarmonic analysis on …definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Fourier transform on locally compact abelian groups concept
  2. Haar measure concept
  3. Locally compact abelian group concept
  4. Locally compact group concept
  5. Unitary representation concept
  6. Harmonic analysis on groups definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Harmonic_analysis
About this page
Tag1GJ6SRT6B7
URIm2://1GJ6SRT6B7
Kinddefinition
WritingW0 stub
MSC43A65, 43A25, 22D10
DefinesHarmonic analysis on groups
UsesLocally compact group, Haar measure, Unitary representation, Locally compact abelian group, Fourier transform on locally compact abelian groups
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.