Hi all, Is there any general way to construct a smooth 2pi periodic function which vanishes on an interval and has positive Fourier coefficients? And if that was too specific I can make this more general by asking if there is any general characterization for functions on the torus with positive Fourier coefficients? Thanks.
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Sure. Let $f$ be a smooth realvalued function supported on $A \subseteq [0,2\pi]$. Then, denoting convolution by $*$, we have:
From 1, if the support of $f$ is a small interval, the support of $f*f$ will be a slightly bigger interval. Clearly, the support of $f$ can be chosen appropriately so that $f*f$ will vanish on any specified interval. 


To make your more general question even more general, on any locally compact abelian group $G$, Bochner's theorem says a continuous function $f$ is the inverse Fourier transform of a positive measure on the dual group if and only if $f$ is positive definite, i.e. $\sum_{n,m=1}^N c_n \overline{c_m} f(x_n  x_m) \ge 0$ for all $c_1 \ldots c_N \in \mathbb C$ and $x_1 \ldots x_N \in G$ 

