Exercise 1, § 1

a) Let $\omega , p, q\in \mathbf{R}$ with $p\leqslant q$. Put $f(x) =e^{i\omega x}$ for $p\leqslant x\leqslant q$, $f(x) = 0$ for $x > q$ or $x < p$. Then for $y\in \mathbf{R}$, one has

b) Let $\omega \in \mathbf{R},\beta >0$. Put $f(x) =e^{(-\beta+i\omega)x}$ for $x\geqslant 0,f(x) = 0$ for $x <0$. Then for $y\in \mathbf{R}$, one has

c) Let $a >0$. Put $f(x) =\frac{1}{a}e^{-a|x|}$ for $x\in \mathbf{R}$. Then for $y\in \mathbf{R}$, one has

d) Put $f(x) =e^{-\pi x^2}$ for $x\in \mathbf{R}$. Then for $y\in \mathbf{R}$, one has $\mathscr{F}(f)(y) =$ $e^{-\pi y^2}$. (Find a first-order linear differential equation satisfied

by the Fourier transform of $f$, and use the formula $\int_0^{+\infty}e^{-t^2}dt=\frac{1}{2}\sqrt{\pi}$ of FVR, VII, §1, No.$^o3.)$

e) Let $n\in \mathbf{N}$ and let Q be a positive definite quadratic form on $\mathbf{R}^n$. Let $\sigma$ be a linear mapping on $\mathbf{R}^n$ such that $Q(x) =\|\sigma (x)\|^2$. Let $Q^*$ be the positive definite quadratic form such that $Q^*(y) =\|^t\sigma^{-1}(y)\|^2$ for every $y\in \mathbf{R}^n$. Let $z\in \mathbf{C}$ be a complex number such that $\mathscr{I}(z)>0$. The mapping $f$ of $\mathbf{R}^n$ into $\mathbf{C}$ defined by $f(x) =$ exp($i\pi zQ(x)$) belongs to $\mathscr{S}(\mathbf{R}^n)$ and one has

for every $y\in \mathbf{R}^n$.

f) Let $f\in \mathscr{C}(\mathbf{R})$ be the function defined by $f(x) =$ (sin($\pi x$)$/(\pi x)$)$^2$ for $x\not= 0$ and $f(0) = 1$. We have $\mathscr{F}(f)(y) = 0$ if $|y|>1$ and $\mathscr{F}(f)(y) = 1- |y|$ otherwise. Let $g(x) =xf(x)$. We have $\mathscr{F}(g)(y) = 0$ if $|y|>1$ and $\mathscr{F}(g)(y) =$ s($y$)$/(2i\pi )$ otherwise.