Let $\Delta_\nu=\{|x_j|<\nu,j=1,...,n\}$ definition of entire function of exponential type $\nu$.
Let $n\in\mathbb{N},~\nu>0 $. A function $g: \mathbb{C}^n \to \mathbb{C} $ is an entire function if the following properties hold
It can be written as power series $~$ $z=(z_1,...,z_n)\in\mathbb{C}^n$: for some $a_{k_1,\dots ,k_n}\in \mathbb{C}$ \begin{equation} g(z)=\sum_{k_1=0}^\infty\dots\sum_{k_n=0}^\infty a_{k_1,\dots ,k_n}z_1^{k_1}\dots z_n^{k_n} \end{equation} for any $z_1,...,z_n\in \mathbb{C}$,\
$\forall\varepsilon>0 \quad\exists A_\varepsilon>0$ such that for all $z\in\mathbb{C}^n$ the following inequality holds
$$ |g(z)| \leq A_\varepsilon e^{({\nu}+\varepsilon)(|z_1|+|z_2|+\dots+|z_n|)}. $$
Let $g\in L_p(\mathbb{R}^n)$
How do I prove that the Fourier transform of entire function g of exponential type $\nu$ is equal to zero outside $\Delta_\nu$?