Exercise 26, § 3

Let $n\in \mathbf{N}$. Endow $\mathbf{R}^n$ with Lebesgue measure $\mu$ and identify $\mathbf{R}^n$ with its dual by the mapping $(x, y)\mapsto$ exp(2$i\pi x\cdot y$).

a) Show that for every function $f\in \mathscr{S}(\mathbf{R}^n)$, the mapping $g\mapsto f*g$ is an automorphism of locally convex spaces of $\mathscr{S}(\mathbf{R}^n)$.

b) Let $f\in \mathscr{S}'(\mathbf{R}^n)$ and $g\in \mathscr{S}(\mathbf{R}^n)$. Show that the mapping $\varphi \mapsto \langle f,\check{g}*\varphi \rangle$ is a tempered distribution, called the convolution of $f$ and $g$. If $f\in \mathscr{S}(\mathbf{R}^n)$, it coincides with the distribution associated with $f*g$. This tempered distribution will again be denoted by $f*g$.

In the questions which follow, fix $f\in \mathscr{S}'(\mathbf{R}^n)$ and $g\in \mathscr{S}(\mathbf{R}^n)$

c) Show that $f*g\in \mathscr{C}^{\infty}(\mathbf{R})$ and

for every $\alpha \in \mathbf{N}^n$.

d) For every $y\in \mathbf{R}^n$, let $g_y$ denote the function $x\mapsto g(y-x)$ on $\mathbf{R}^n$. Show that the function $f*g$ satisfies

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

e) Let $h\in \mathscr{S}(\mathbf{R}^n)$. Show that $(f*g)*h=f*(g*h)$.

f ) Show that the function $f*g$ has polynomial growth, as do all its derivatives. (Use exercise 25.)

g) Show that $\mathscr{F}(f*g) =\mathscr{F}(f)\mathscr{F}(g)$.

h) Show that the Fourier transform of $f g$ is an infinitely differentiable function on $\mathbf{R}^n$ and has polynomial growth, as do all its derivatives. Moreover, for every $y\in \mathbf{R}^n$, one has

where $e_y$ denotes the character $x\mapsto$ exp(2$i\pi x\cdot y$) of $\mathbf{R}^n$.