Fourier transform on a locally compact abelian group
= Fourier transform on a locally compact abelian group
{c}
{title2=$\widehat f(\chi)=\int_G f(x)\overline{\chi(x)}\,dm_G(x)$}
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.