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Let $f $ be a periodic function and denote by $c_n$, for $n \in \mathbb{N}$, its Fourier coefficients, i.e. $$ c_n := \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{inx}\ dx. $$ It is well known that Bochner's theorem states that the Fourier transform of a positive definite function is positive. Does this result extend also to Fourier coefficients? That is, if $f$ is positive definite on say $[-\pi,\pi]$, does this imply the positiveness of the Fourier coefficients $c_n$? (Possibly excluding $n = 0$).

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    $\begingroup$ Yes. As explained in the WIkipedia page you quote, Bochner's theorem holds on any locally compact abelian group, in particular $\mathbb{R}/\mathbb{Z}$. $\endgroup$
    – abx
    Mar 1, 2021 at 17:59

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Following from the comment made by @abx, and some further research. The answer is yes, and this statement is often called Herglotz's Theorem. I found a useful resource regarding this here.

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