= Fourier coefficient of a measure on the circle
{c}
{title2=$\widehat\mu(n)=\int e^{-inx}\,d\mu(x)$}
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 $|\widehat\mu(n)|\leq\|\mu\|$, 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.
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