 Fourier transform on a locally compact abelian group

For $f\in L^1(G)$ the displayed integral defines a bounded continuous function on the Pontryagin dual group. It takes convolution to pointwise multiplication and left translation $T_x$ to multiplication by $\overline{\chi (x)}$. The transform depends on the scale of Haar measure on $G$; the compatible dual Haar measure scales reciprocally.