Fourier transform on locally compact abelian groups

Summary

The Fourier transform on a locally compact abelian group $G$ sends an integrable function $f$ on $G$ to a function $\widehat{f}$ on the dual group. The value of $\widehat{f}$ at a character is the integral of $f$ against the conjugate of that character. It unifies Fourier series on the circle, the Fourier transform on $\R^n$ and the discrete Fourier transform on $\Z/n\Z$. It turns convolution into pointwise multiplication, and with the right normalisation it is unitary on $L^2$.