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Convolutions on groups

If G is a suitable group endowed with a measure λ, and if f and g are real or complex valued integrable functions on G, then we can define their convolution by

It is not commutative in general. In typical cases of interest G is a locally compact Hausdorff topological group and λ is a (left-) Haar measure. In that case, unless G is unimodular, the convolution defined in this way is not the same as ∫ f (x y − 1) g (y) d λ (y) {\textstyle \int f\left(xy^{-1}\right)g(y)\,d\lambda (y)}. The preference of one over the other is made so that convolution with a fixed function g commutes with left translation in the group:

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 groups, 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 (T), we have the following familiar operator acting on the Hilbert space L (T):

The operator T is compact. A direct calculation shows that its adjoint T* is convolution with

By the commutativity property cited above, T is 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 spectral theory, there exists an orthonormal basis { h } that simultaneously diagonalizes S. This characterizes convolutions on the circle. Specifically, we have

which are precisely the characters 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 representations form an orthonormal basis in L 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.