Here is a few transform that I'm trying to calculate:

1. Calculate the fourier coefficient of $\displaystyle e^x$ on $\displaystyle [-\pi, \pi]$ and thus determine the sum $\displaystyle \sum_{k=-\infty}^{\infty} 1/(1+k^2)$. I get $\displaystyle F_k = (-1)^k (e^{\pi}-e^{-\pi})(1+ki) /(1+k^2)$. I'm not sure if this is correct and can't see how using any identities (e.g. Parseval) I can calculate the sum.

2. Suppose that $\displaystyle f \in \{C(\mathbb{R}) \cap L^1 (\mathbb{R})\}$ and $\displaystyle p \in \mathbb{N} $. Show that if $\displaystyle |k|^p F_k $ is integrable then $\displaystyle f$ has $\displaystyle p$ continuous bounded derivatives.