Fourier transform on locally compact abelian groups

Remarks

Example: for $G = \R$ with Lebesgue measure, the continuous characters are $\chi_\xi(x) = e^{2\pi i \xi x}$ for $\xi \in \R$, and the formula gives $\widehat{f}(\chi_\xi) = \int_\R f(x) e^{-2\pi i \xi x} \, dx$, the classical Fourier transform. For $G = \mathbb{T}$ with normalised arc length and $\chi_n(z) = z^n$, it gives the Fourier coefficients of $f$. Non-example: for $f(x) = 1$ on $\R$, the integral does not converge because $f \notin L^1(\R)$; the transform of such $f$ only makes sense as a distribution. The integral converges for $f \in L^1$ because $\abs{\chi(x)} = 1$, and $\abs{\widehat{f}(\chi)} \le \norm{f}_1$. The function $\widehat{f}$ is continuous on $\widehat{G}$ and vanishes at infinity, and $\widehat{f * g} = \widehat{f}\,\widehat{g}$ for the convolution $f * g$. Some authors use $\chi(x)$ instead of $\overline{\chi(x)}$; this changes $\widehat{f}(\chi)$ into $\widehat{f}(\overline{\chi})$.