Editing Harmonic analysis (section)

==Abstract harmonic analysis==
Abstract harmonic analysis is primarily concerned with how real or
complex-valued [[function (mathematics)|function]]s (often on very general domains) can be studied using symmetries such
as [[translations]] or [[rotations]] (for instance via the [[Fourier transform]] and its relatives); this field is of
course related to real-variable harmonic analysis, but is perhaps closer in spirit to [[representation theory]] and [[functional analysis]].<ref name=tao/>

One of the most modern branches of harmonic analysis, having its roots in the mid-20th century, is [[mathematical analysis|analysis]] on [[topological group]]s. The core motivating ideas are the various [[Fourier transform]]s, which can be generalized to a transform of functions defined on Hausdorff [[locally compact group|locally compact topological groups]].<ref>{{cite book|title=Introduction to the Representation Theory of Compact and Locally Compact Groups|author=Alain Robert}}</ref>

One of the major results in the theory of functions on [[abelian group|abelian]] locally compact groups is called [[Pontryagin duality]]. 
Harmonic analysis studies the properties of that duality. Different generalization of  Fourier transforms attempts to extend those features to different settings, for instance, first to the case of general abelian topological groups and second to the case of non-abelian [[Lie group]]s.<ref>{{cite book|title=A Course in Abstract Harmonic Analysis|author=Gerald B Folland}}</ref>

Harmonic analysis is closely related to the theory of unitary group representations for general non-abelian locally compact groups. For compact groups, the [[Peter–Weyl theorem]] explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.<ref>{{cite book|title=Introduction to the Representation Theory of Compact and Locally Compact Groups|author=Alain Robert}}</ref> This choice of harmonics enjoys some of the valuable properties of the classical Fourier transform in terms of carrying convolutions to pointwise products or otherwise showing a certain understanding of the underlying [[Group (mathematics)|group]] structure. See also: [[Non-commutative harmonic analysis]].

If the group is neither abelian nor compact, no general satisfactory theory is currently known ("satisfactory" means at least as strong as the [[Plancherel theorem]]). However, many specific cases have been analyzed, for example, [[Special linear group|SL<sub>''n''</sub>]]. In this case, [[Group representation|representations]] in infinite [[Dimension (mathematics and physics)|dimensions]] play a crucial role.