Fourier coefficient of a measure on the circle ($\hat{\mu }(n)=\int e^{-inx}d\mu (x)$)

For a finite complex measure on the additive circle group $\mathbb{R}/(2\pi \mathbb{Z})$, its Fourier coefficients are integrals against the group characters $e^{-inx}$, $n\in \mathbb{Z}$. They obey $|\hat{\mu }(n)|\le \Vert \mu \Vert$, where the norm is the total variation norm of a measure. A probability measure has zeroth coefficient one. The normalized Haar measure $m(dx)=dx/(2\pi )$ has all nonzero-frequency coefficients zero. A bounded family of finite Radon measures converges in the weak-star topology exactly when its coefficients converge to those of the proposed limit: extend linear combinations to all continuous tests using the density of trigonometric polynomials and the common variation bound.