Editing Fourier series (section)

=== Compact groups ===
{{main|Compact group|Lie group|Peter–Weyl theorem}}

One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any [[compact group]]. Typical examples include those [[classical group]]s that are compact. This generalizes the Fourier transform to all spaces of the form ''L''<sup>2</sup>(''G''), where ''G'' is a compact group, in such a way that the Fourier transform carries [[convolution]]s to pointwise products. The Fourier series exists and converges in similar ways to the {{closed-closed|−''π'',''π''}} case.

An alternative extension to compact groups is the [[Peter–Weyl theorem]], which proves results about representations of compact groups analogous to those about finite groups.

[[File:F orbital.png|thumb|right|The [[atomic orbital]]s of [[chemistry]] are partially described by [[spherical harmonic]]s, which can be used to produce Fourier series on the [[sphere]].]]