Harmonic analysis

The classical Fourier transform is still an interesting area of research. For instance, if we impose some requirements on a function f, we can attempt to translate these requirements in terms of the Fourier transform of f. For example, if a function is compactly supported, then its Fourier transform may not also be compactly supported; this is a very elementary form of an Uncertainty Principle in a Harmonic Analysis setting (there are more sophisticated examples of this.)

Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional 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 locally compact 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. It is developed in detail on its dedicated page.

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.

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.

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