Fourier series on a compact group

Remarks

The term for $\pi$ equals $d_\pi (f * \chi_\pi)(g)$, where $\chi_\pi = \tr \pi$ is the character and $*$ is convolution. It is the orthogonal projection of $f$ onto the span of the matrix coefficients of $\pi$. For $f \in L^2(G)$ the series converges to $f$ in $L^2(G)$, and only countably many terms are nonzero; both facts follow from the Peter–Weyl theorem. Convergence may fail pointwise and even in the uniform norm for continuous $f$, as already on the circle.

Example: let $G = \R / 2\pi\Z$ with $dg = d\theta / 2\pi$. The irreducible unitary representations are $\pi_n(\theta) = e^{i n \theta}$ for $n \in \Z$, all with $d_\pi = 1$. Then $\hat f(\pi_n) = \frac{1}{2\pi} \int_0^{2\pi} f(\theta) e^{-i n \theta} \, d\theta$, and the Fourier series is $\sum_n \hat f(\pi_n) e^{i n \theta}$, the classical one.

Non-example: the sum $\sum_\pi \tr(\hat f(\pi) \pi(g))$ without the factors $d_\pi$ is not the Fourier series. For $f = \chi_\sigma$ with $d_\sigma > 1$, Schur orthogonality gives $\hat f(\pi) = 0$ for $\pi \ne \sigma$ and $\hat f(\sigma) = d_\sigma^{-1} \, \mathrm{id}$, so that sum equals $d_\sigma^{-1} \chi_\sigma \ne f$.