= Bochner–Herglotz theorem
{c}
{title2=$r(n)=\int_{\mathbb R/\mathbb Z}e^{2\pi i nt}\,d\sigma(t)$}
= Bochner-Herglotz spectral theorem
{c}
{synonym}
A <positive-definite function> $r$ on the discrete group $\mathbb Z$ is the sequence of <Fourier coefficients> of a unique finite positive <Borel measure> on the circle, with total mass $r(0)$. Positivity makes the finite sums $\sum_{|n|<N}(1-|n|/N)r(n)e^{-2\pi int}$ nonnegative. Their measures converge weakly to the required measure; uniqueness follows from density of <trigonometric polynomials> in continuous functions.
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