Let $v:\mathbb{C}\rightarrow \mathbb{C}$ be entire function with an estimate $|v(z)| \leq C_N(1+|z|)^{-N}e^{R|Imz|}$ for all $z \in \mathbb{C}$ then there is $u \in S(\mathbb{R})$ such that $supp(u) \subset B(0,R)$ with $\hat{u} = v$.
By fourier-inversion formula we have that $u(x) = \int_{\mathbb{R}}v(\varepsilon)e^{i\varepsilon x}d\varepsilon$ so we have existence. Now we need to show that $u$ is in $S(\mathbb{R})$ and that $supp(u) \subset B(0,R)$.
Im struggling to find an estimate for $|\partial^N(u)| < \infty$. I tried to apply Cauchy’s local integral formula, but i end up with a mess... Also i tried to apply the estimate to $\partial_N(u)$, but then the integral diverges.
We can easily compute $\partial^N(u)$ with respect to $x$ which is $i^N\int_{\mathbb{R}}\varepsilon^N v(\varepsilon)e^{i\varepsilon x}d\varepsilon$. If we now apply the estimate to the $|\partial^N(u)|$ the integral diverges.
Any suggestions? Thanks.