MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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.

share|cite|improve this answer
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 $x_1 \ldots x_N \in G$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.