Convolution on a locally compact group

Let $G$ be a locally compact group with a fixed left Haar measure $\mu$ . For functions $f,g:G\to\mathbb C$ for which the integral exists, their convolution is

(f*g)(x) = \int_G f(y)g(y^{-1}x)\,d\mu(y).

In particular, this formula defines $f*g$ for compactly supported continuous functions and, up to almost-everywhere equality, for $f,g\in L^1(G)$ . The order of the factors matters: unless $G$ is abelian, $f*g$ need not equal $g*f$ . Fixing left rather than right Haar measure also fixes the displayed convention.

Algebraic and analytic properties

Convolution is bilinear and associative whenever the relevant integrals are defined. On $L^1(G)$ , it satisfies

\lVert f*g\rVert_1\leq \lVert f\rVert_1\lVert g\rVert_1.

Consequently $L^1(G)$ becomes a Banach algebra under convolution. Compactly supported continuous functions are also closed under convolution. The proofs combine left invariance of Haar measure with Tonelli’s theorem and a change of variables.

Translation interpretation

Writing $L_y g(x)=g(y^{-1}x)$ , the value $(f*g)(x)$ averages the left translates $L_y g$ , weighted by $f(y)$ . This viewpoint explains why convolution converts group representations into operators: a representation can integrate the operators assigned to group elements against $f$ .

Involution and modular correction

For non-unimodular groups, inversion does not preserve a left Haar measure. The natural star operation on the convolution algebra must therefore include the modular function. The precise correction is recorded in the involution on a group convolution algebra; omitting it generally breaks the identity $(f*g)^*=g^{*}*f^{*}$ .

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. Routledge publisher record. Relevant: Chapter 2, “Convolutions.”
Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Volume I, Springer, 1963. Springer DOI record. Relevant: “Convolutions and Group Representations.”

Euclidean and one-sided conventions

On the additive group $\mathbb R^n$ , Lebesgue measure gives

(f*g)(x)=\int_{\mathbb R^n}f(y)g(x-y)\,dy.

For functions on the positive time axis extended by zero to negative times, the same formula becomes $(f*g)(t)=\int_0^t f(s)g(t-s)\,ds$ for $t>0$ . The choice of zero extension is part of this one-sided convention.