Fourier transform on a locally compact abelian group ($\hat{f}(\chi )={\int }_{G}f(x)\overline{\chi (x)}d{m}_G(x)$)

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.

For the dual Haar measure, the formula first follows for continuous compactly supported convolution squares from the Bochner theorem. More generally, if $f\in L^1(G)$ and $\hat{f}\in L^1(\hat{G})$, the right-hand side is a continuous representative of $f$; equality holds everywhere if $f$ is continuous, and otherwise at every point where the chosen representative is continuous. Convolution with normalized small-support positive convolution squares proves the extension: the spatial convolutions approach $f$ in $L^1$, while their transforms approach $\hat{f}$ in $L^1$ by equicontinuity of compact families of characters and an integrable-tail split.