Abstract Harmonic Analysis
One of the more modern branches of harmonic analysis, having its roots in the mid-twentieth century, is analysis on topological groups. The core motivating idea are the various Fourier transforms, which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups.
The theory for abelian locally compact groups is called Pontryagin duality; it is considered to be in a satisfactory state, as far as explaining the main features of harmonic analysis goes.
Harmonic analysis studies the properties of that duality and Fourier transform; and attempts to extend those features to different settings, for instance to the case of non-abelian Lie groups.
For general nonabelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations. 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. This choice of harmonics enjoys some of the useful properties of the classical Fourier transform in terms of carrying convolutions to pointwise products, or otherwise showing a certain understanding of the underlying group structure. See also: Non-commutative harmonic analysis.
If the group is neither abelian nor compact, no general satisfactory theory is currently known. By "satisfactory" one would mean at least the equivalent of Plancherel theorem. However, many specific cases have been analyzed, for example SLn. In this case, representations in infinite dimension play a crucial role.
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