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.



Positive Fourier coefficients
Sure. Let $f$ be a smooth real-valued function supported on $A \subseteq [0,2\pi]$. Then, denoting convolution by $\*$, we have:


1. f\*f s supported on the Minkowski sum $A+A$
2. f\*f is smooth
3. $\widehat{f\*f}(n) = |\hat{f}(n)|^2 \geq 0$.


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. 


