Plancherel theorem: Difference between revisions

Revision as of 12:17, 11 January 2025

Revision as of 05:30, 29 January 2025

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==Locally compact groups==

There is also a Plancherel theorem for the Fourier transform in [[locally compact group]]s. In the case of an [[abelian group]] <math>G</math>, there is a [[PontrjaginPontryagin dual]] group <math>\widehat G</math> of characters on <math>G</math>. Given a [[Haar measure]] on <math>G</math>, the Fourier transform of a function in <math>L^1(G)</math> is

<math display="block">\hat f(\chi) = \int_G \overline{\chi(g)}f(g)\,dg</math>

for <math>\chi</math> a character on <math>G</math>.