Fourier basis on a compact abelian group

Statement

Let $K$ be a compact metrizable abelian group with normalized Haar measure $m$ . Its continuous unitary characters $\widehat K$ form an orthonormal basis of $L^2(K,m)$ .

Orthogonality

If $\chi$ is nontrivial, choose $a$ with $\chi(a)\ne1$ . Translation invariance gives $\int\chi\,dm=\chi(a)\int\chi\,dm$ , so the integral is zero. Apply this to $\chi\overline\psi$ to get orthogonality; each character has norm one.

Completeness and torus coordinates

Characters separate points; their finite linear combinations form a conjugation-stable algebra containing constants. The Stone–Weierstrass theorem gives density in continuous functions, hence in $L^2$ . For $K=\mathbb R^d/\mathbb Z^d$ , the characters are $e_k(x)=e^{2\pi i k\cdot x}$ , $k\in\mathbb Z^d$ .

This is the compact-group form of Plancherel theory. It permits fixed functions to be tested one Fourier coefficient at a time.