I want to find a Schwartz function $\varphi$ with the following properties \begin{equation} \varphi > 0 \quad \text{and} \quad \mathrm{supp}(\widehat{\varphi})\subset B_{R}, \end{equation} where $B_{R}$ is a ball with radius R.
My first thought was something like $\varphi = g^{2}$ , because then $\widehat{\varphi} = \widehat{g}\ast\widehat{g}$. If $g$ is then a function where the Fourier Transform has $B_{\frac{R}{2}}$ , then $\mathrm{supp}(\widehat{g}\ast\widehat{g})\subset B_{R}$ but $g^{2}\geq 0$. So this don't work. My next thought was to take the Gaussian Function and convolve it with a Schwartz function $f$ with compactly supported Fourier Transform. I know $f$ is analytic. But I'm not sure if the convolution from $f$ with the Gaussian is strictly positive?