Fourier transforms of exponentially decaying functions - analyticity in a strip

Suppose there exists a positive number $a$ such that $e^{a\left|x\right|}. f \in L_{1}(\mathbb {R})$. Prove that $\hat{f}(y)$ extends to a function $\tilde{f}(z)$, analytic in the strip $\left|\operatorname {Im} z\right| < a$.

Let a function $f \in L_{2}(\mathbb {R})$ be almost everywhere nonzero and suppose there exists a positive number $a$ such that $e^{a\left|x\right|} \cdot f \in L_{2}(\mathbb {R})$. Prove that the system of functions $\left\{ x^{n} f\right\}_{n = 0}^{\infty }$ is complete in $L_{2}(\mathbb {R})$.

Prove the statement of problem 3.54 using the Fourier transform.

Let $f\in L^{1}\left({\mathbb {R},dx}\right)$ be a rational function. Prove that $\left|\tilde f(\lambda )\right|\leq ce^{-\varepsilon \left|\lambda \right|}$, $\lambda \in \mathbb {R}$, for some constants $c>0$ and $\varepsilon >0$.