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  "body": "\n## Definition\n\n*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:\n\n1. how functions in spaces such as $L^1(G)$ and $L^2(G)$ decompose into components on which $G$ acts irreducibly;\n2. how the [[unitary representations|74BPYJNZTM]] of $G$, in particular the regular representation on $L^2(G)$, decompose into irreducible ones;\n3. how translation, convolution and invariant differential operators act on these components.\n\n## Remarks\n\nExample: for a [[locally compact abelian group|ZR09S9R3AW]] $G$, the irreducible unitary representations are the continuous characters.\nThe decomposition is then given by the [[Fourier transform|VHN5VW5Q92]] from functions on $G$ to functions on the dual group.\nFor $G = \\mathbb{T}$ this is the theory of Fourier series.\nFor compact groups the decomposition is a direct sum over finite-dimensional irreducible representations, by the Peter–Weyl theorem.\nNon-example: the study of the zeros of a single polynomial on $\\R$ uses no group action, and it is not harmonic analysis on groups.\nThe terms abstract harmonic analysis and, for non-abelian $G$, non-commutative harmonic analysis are also used.\n",
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