# Nonnegative Fourier series coefficients for periodic nonnegative-definite function

Is there a simple way to show that the Fourier series coefficients of a periodic, nonnegative-definite function $$\kappa$$ must all be nonnegative? (By nonnegative-definite I mean that the Gram matrix $$\Sigma_{ij}=\kappa(x_i-x_j)$$ is nonnegative-definite for any sequence of real numbers $$x_1,\ldots,x_n$$. AKA positive semi-definite.)

It seems that some version of Bochner's Theorem ought to do it, but I'm having trouble finding the version appropriate to periodic functions.