Editing Convolution (section)

== Convolutions on groups ==
If ''G'' is a suitable [[group (mathematics)|group]] endowed with a [[measure (mathematics)|measure]] λ, and if ''f'' and ''g'' are real or complex valued [[Lebesgue integral|integrable]] functions on ''G'', then we can define their convolution by

:<math>(f * g)(x) = \int_G f(y) g\left(y^{-1}x\right)\,d\lambda(y).</math>

It is not commutative in general. In typical cases of interest ''G'' is a [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] and λ is a (left-) [[Haar measure]]. In that case, unless ''G'' is [[unimodular group|unimodular]], the convolution defined in this way is not the same as <math display="inline">\int f\left(xy^{-1}\right)g(y) \, d\lambda(y)</math>. The preference of one over the other is made so that convolution with a fixed function ''g'' commutes with left translation in the group:

:<math>L_h(f* g) = (L_hf)* g.</math>

Furthermore, the convention is also required for consistency with the definition of the convolution of measures given below. However, with a right instead of a left Haar measure, the latter integral is preferred over the former.

On [[locally compact abelian group]]s, a version of the [[convolution theorem]] holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. The [[circle group]] '''T''' with the Lebesgue measure is an immediate example. For a fixed ''g'' in ''L''<sup>1</sup>('''T'''), we have the following familiar operator acting on the [[Hilbert space]] ''L''<sup>2</sup>('''T'''):

:<math>T {f}(x) = \frac{1}{2 \pi} \int_{\mathbf{T}} {f}(y) g( x - y) \, dy.</math>

The operator ''T'' is [[compact operator on Hilbert space|compact]]. A direct calculation shows that its adjoint ''T* '' is convolution with

:<math>\bar{g}(-y).</math>

By the commutativity property cited above, ''T'' is [[normal operator|normal]]: ''T''* ''T'' = ''TT''* . Also, ''T'' commutes with the translation operators. Consider the family ''S'' of operators consisting of all such convolutions and the translation operators. Then ''S'' is a commuting family of normal operators. According to [[compact operator on Hilbert space|spectral theory]], there exists an orthonormal basis {''h<sub>k</sub>''} that simultaneously diagonalizes ''S''. This characterizes convolutions on the circle. Specifically, we have

:<math>h_k (x) = e^{ikx}, \quad k \in \mathbb{Z},\;</math>

which are precisely the [[Character (mathematics)|character]]s of '''T'''. Each convolution is a compact [[multiplication operator]] in this basis. This can be viewed as a version of the convolution theorem discussed above.

A discrete example is a finite [[cyclic group]] of order ''n''. Convolution operators are here represented by [[circulant matrices]], and can be diagonalized by the [[discrete Fourier transform]].

A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional [[unitary representation]]s form an orthonormal basis in ''L''<sup>2</sup> by the [[Peter–Weyl theorem]], and an analog of the convolution theorem continues to hold, along with many other aspects of [[harmonic analysis]] that depend on the Fourier transform.