Test functions with small support and nonnegative Fourier transform - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:45:37Z http://mathoverflow.net/feeds/question/110881 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110881/test-functions-with-small-support-and-nonnegative-fourier-transform Test functions with small support and nonnegative Fourier transform km 2012-10-28T06:38:49Z 2012-10-28T15:31:33Z <p>The following problem arose in <a href="http://mathoverflow.net/questions/110427/the-identity-element-of-a-compact-group-is-a-limit-point-of-any-polynomial-seque" rel="nofollow">a question</a> I recently asked : given a (possibly non abelian) compact group $G$ and a neighbourhood $U$ of the identity in $G$, can we always find a function $f : G \mapsto \mathbb{R}$, which vanishes outside $U$, whose Fourier transform is nonnegative, and which satisfies $\hat f(1) \neq 0$ ?</p> http://mathoverflow.net/questions/110881/test-functions-with-small-support-and-nonnegative-fourier-transform/110909#110909 Answer by Abdelmalek Abdesselam for Test functions with small support and nonnegative Fourier transform Abdelmalek Abdesselam 2012-10-28T15:31:33Z 2012-10-28T15:31:33Z <p>I didn't think about the non Abelian case, but for a commutative group say like $\mathbb{R}$ you can do the following. Take a function with support in $\frac{1}{2} U$ and symmetric with respect to the origin. Its convolution with itself answers your question.</p>