# To find a positive function with compact spectrum

Let $e_1=(0,1)^T$, $$S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},$$ is a cone in $\mathbb{R}^2$.

I want to find a non-trivial smooth function which satisfies: $$f\geq 0; \ \ f\equiv0 \ \mbox{on} \ S^c; \ \ \mbox{supp}\ \hat{f}\subset B_1(0).$$ Here $\hat{f}$ is the Fourier transform of $f$. I found using so called "real paley wiener theorem", which was proved in "Andersen, Nils, and Marcel de Jeu. Real Paley-Wiener theorems and local spectral radius formulas. Transactions of the American Mathematical Society 362.7 (2010): 3613-3640," the last condition can be rewrote as:
$$|\nabla^k f(x)|\leq Ck^N(1+|x|)^N.$$

Thanks a lot if one can give me any comment or reference.

• What is $\hat f$? Fourier transform of $f$? If so, such a function does not exist. If the Fourier transform has compact support, the function is analytic, so it cannot vanish on the complement of $S$ unless it is identically zero. – Michael Renardy May 11 '17 at 15:50
• Yes, $\hat{f}$ is the Fourier transform of $f$. You mean a real analytic function cannot vanish on an open set in $\mathbb{R}^2$? – John Zhao May 11 '17 at 22:36
• @JohnZhao: yes, as explained here – Nik Weaver May 12 '17 at 2:46
• @MichaelRenardy, Oh, I see. My question is really trivial! Thank you so much for your help. – John Zhao May 12 '17 at 13:47