Fourier convolution theorem on a locally compact abelian group

Statement

Let $G$ be a locally compact group that is abelian, with a fixed Haar measure. For $f,g\in L^1(G)$ , define their convolution using that measure. The Fourier convolution theorem states that

\widehat{f*g}(\gamma)=\widehat f(\gamma)\widehat g(\gamma) \qquad(\gamma\in\widehat G),

where hats denote the Fourier transform on $G$ and $\widehat G$ is the Pontryagin dual. Thus the Fourier transform is an algebra homomorphism from the convolution algebra $L^1(G)$ to continuous functions on $\widehat G$ with pointwise multiplication.

Proof mechanism

Expanding the definitions gives

\widehat{f*g}(\gamma) =\int_G\int_G f(y)g(y^{-1}x)\overline{\gamma(x)}\,dy\,dx.

Absolute integrability permits Fubini's theorem. The substitution $x=yz$ , left invariance of Haar measure, and $\gamma(yz)=\gamma(y)\gamma(z)$ split the double integral into $\widehat f(\gamma)\widehat g(\gamma)$ . This calculation also explains exactly where commutativity enters: it makes scalar characters sufficient to diagonalize convolution Folland, Chapter 4.

Analytic consequences

Convolution operators become multiplication operators after Fourier transformation. With the dual Haar measure, the Plancherel theorem therefore converts suitable translation-invariant operators on $L^2(G)$ into multiplication by Fourier multipliers. The identity also shows that zeros of Fourier transforms control invertibility and ideal structure in the commutative $L^1$ group algebra.

Examples and conventions

For $G=\mathbb R^n$ , this is the classical identity $\mathcal F(f*g)=(\mathcal Ff)(\mathcal Fg)$ . For $G=\mathbb Z$ , convolution of sequences becomes multiplication of Fourier series on the circle. Changing the sign or $2\pi$ convention in both transforms leaves the product identity unchanged.

References

W. Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. DOI record. Relevant: Fourier transforms and convolution on locally compact abelian groups.
G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 4, Fourier analysis on locally compact abelian groups.