It's fairly straightforward to prove the following, which seem like special cases of Bochner's Theorem for periodic functions, but don't quite match the statements of Bochner's Theorem that I've seen, because they give a specific form for the nonnegative measure. (To my knowledge Bochner's just states that there exists a certain nonnegative measure.) Are the following stated and proven somewhere in the literature? I've Googled around, but haven't found anything.
- Let $f:\mathbb{Z}\to\mathbb{R}$ be an even function with integer period $s$. Then $f$ is positive semidefinite if and only if the discrete Fourier transform $c_0,\ldots,c_{s-1}$ of the sequence $f(0),\ldots,f(s-1)$ satisfies $c_k\geq 0$ for all $k$.
- Let $f:\mathbb{R}\to\mathbb{R}$ be an even function with period $s$ whose Fourier series converge on $[0,s]$. Then $f$ is positive semidefinite if and only if its Fourier series coefficients are all nonnegative.
In (2) above I refer to the exponential form of the Fourier series, i.e. the Fourier series coefficients $c_k$ satisfy
$$ f(x) = \sum_{k=-\infty}^{\infty} c_k \exp(\imath 2 \pi k x / s). $$