Fourier series on a compact group

Summary

The Fourier series of a square integrable function on a compact group expands it along the irreducible unitary representations of the group. Its coefficients are operators, one for each irreducible representation, and its terms are combinations of matrix coefficients. By the Peter–Weyl theorem the series converges to the function in $L^2$. For the circle group it is the classical Fourier series.