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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.

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