Let $f \in C_c^\infty(\mathbb{R}^n)$. Then, $\hat{f}(\xi)= (2\pi)^{\frac{-n}{2}}\int_{\mathbb{R^n}}\exp(-ix\xi)f(x){d}x$
can be (i) analytically continued to an entire function and (ii) for $r>0$ there holds: $\hat{f}(\cdot + ib)$ is uniformly rapidly decreasing $\forall\; b\in B(0,r)$.
Can somebody help me to see (i) and (ii)?
Thanks.