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A strongly continuous unitary representation of the circle group splits as an orthogonal sum of character subspaces. The projections are $P_n={\int }_{\mathbb{T}}{z}^{-n}{U}_{z}dz$. Character orthogonality makes them mutually orthogonal, and Fejér sum shows their ranges span the represented Hilbert space. The self-adjoint generator has integer eigenvalues, with domain given by ${\sum }_n{n}^2\Vert P_n\xi {\Vert }^2<\infty$.

If a periodic integrable function has all Fourier coefficients zero, it vanishes almost everywhere. Its Fejér sums vanish, whereas the Fejér kernel is an approximate identity and converges to that function in the integrable norm. Applied to a periodized squared Fourier transform, this detects orthogonality of integer translates.