# Does Bochner's Theorem apply to Fourier coefficients?

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 $$n = 0$$).

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