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](https://en.wikipedia.org/wiki/Bochner%27s_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 $n = 0$).



Does Bochner's Theorem apply to Fourier coefficients?
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](https://www.orsj.or.jp/%7Earchive/pdf/e_mag/Vol.60_02_122.pdf).


