$f$ is a piecewise continuous real function. There are $a>0$ and $C>0$, so that $|f(x)|<Ce^{-a|x|}$ for all $x\in\mathbb{R}$. I want to show that Fourier transformation $\hat{f}$ is regular in $|\text{Im }\zeta|<a$.
Second question: $f$ is regular in $|\text{Im }z|<a$ for some $a>0$ and continuous to its boundary. Assume that for all $y\in [-a,a]$ $$\int_{-\infty}^\infty |f(x+iy)|dx<\infty$$ and $$\lim_{|x|\to \infty}\max_{y\in [-a,a]}|f(x+iy)|=0.$$ I want to show that there are $C>0$ so that Fourier transformation $\hat{f}$ satisfies $|\hat{f}(s)|\leqslant Ce^{-a|s|}$ for all $s\in \mathbb{R}$.
Any ideas on how to approach these problems? I though about using Paley-Wiener theorems. Am I on the right track?
First question: Simply define $$F(z)=\int f(t)e^{-itz}\,dt\quad(|\Im z|<a)$$ and use the growth condition to show $F$ is holomorphic. (For example, the growth condition justifies differentiating under the integral sign. Or, more fun: Use dominated convergence to show $F$ is continuous, and then Morera+Fubini to show $F$ is holomorphic; the growth condition implies the technicality needed to apply Fubini.)
Second question: Let $$f_y(x)=f(x+iy)\quad(|y|\le a).$$Use Cauchy's Theorem to show that $$\hat f_y(\xi)=e^{\pm y\xi}\hat f(\xi).$$When you figure out which it is come back to this post and replace the $\pm$ by $+$ or $-$, whichever one's right. (Apply CT to the obvious rectangle; the second hypothesis is enough to show that the integrals over the vertical edges tend to $0$ as the base tends to $(-\infty,\infty)$.)
Now $f_a,f_{-a}\in L^1$ shows that $\hat f_a$ and $\hat f_{-a}$ are bounded, which gives the inequality you want.