Calculate the Fourier transform of the following functions/measures on \(\mathbb{R}\) :

(a) \(\mathbb{1}_{[-1,1]}(x)\),

(b) \(\mathbb{1}_{[-1,1]} \star \mathbb{1}_{[-1,1]}(x)\),

(c) \(e^{-x} \mathbb{1}_{[0, \infty)}(x)\),

(d) \(e^{-|x|}\),

(e) \(1 /\left(1+x^{2}ight)\),

(f) \(\quad(1-|x|) \mathbb{1}_{[-1,1]}(x)\),

(g) \(\sum_{k=0}^{\infty} \frac{t^{k}}{k !} e^{-t} \delta_{k}\)

(h) \(\sum_{k=0}^{n}\left(\begin{array}{l}n \\ k\end{array}ight) p^{k} q^{n-k} \delta_{k}\)

\footnotetext{

\({ }^{1}\) With some effort one can make this explicit using the Leibniz formula for derivatives of products.

\({ }^{2}\) For clarity, we use \(\mathscr{F}_{x ightarrow \xi}[u(x)](\xi)\) to denote \(\mathscr{F} u(\xi)=\widehat{u}(\xi)\).

}