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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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2 Answers 2

up vote 6 down vote accepted

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.

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3  
Minor point --- don't you want $f * g$ where $g(t) = \bar{f}(-t)$? The Fourier coefficients of $f * f$ are $(\hat{f}(n))^2$, not $|\hat{f}(n)|^2$. –  Nik Weaver Jun 13 '12 at 19:52

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 $x_1 \ldots x_N \in G$

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