Exercise 3, § 1

a) The functions $t\mapsto t^ne^{-\pi t^2}$ for $n$ integer $\geqslant 0$ form a total set in $L^2(\mathbf{R})$. (Let $f\in L^2(\mathbf{R})$ be an element orthogonal to the $t^ne^{-\pi t^2}$. For $z\in \mathbf{C}$, set $F(z) =\int_{-\infty}^{+\infty}f(t)e^{-\pi t^2}e^{2i\pi tz}dt$, and now prove that F is an entire function whose derivatives at 0 are all zero. Deduce that the Fourier transform of the function $t\mapsto f(t)e^{-\pi t^2}$ on $\mathbf{R}$ is zero.)

b) For $k\in \mathbf{N}$, one has $\partial_t^k(e^{-2\pi t^2}) = P_k(t)e^{-2\pi t^2}$, where $P_k$ is a polynomial of degree $k$, with leading coefficient $(-4\pi )^k$.

c) Let $k\in \mathbf{N},P\in \mathbf{C}[t]$ nonzero and $\alpha$ the leading coefficient of P. Then

$\int_{-2\pi t^2}\{0$ if deg(P) $< k$

$_{\mathbf{R}}(-1)^kk$!2$^{-1/2}\alpha$ if deg(P) $=k$.

d) For $k\in \mathbf{N}$ and $t\in \mathbf{R}$, set :

$\mathscr{H}_k(t) = (-1)^k2\surd \overset{1/4k-}{\pi}k^k$! $e^{\pi t^2}\partial^k_t(e^{-2\pi t^2})$

(“Hermite functions”). The family $(\mathscr{H}_k)$ is an orthonormal basis of the space $L^2(\mathbf{R})$. (Use a), b), c)).

e) One has $\mathscr{F}(\mathscr{H}_k) = (-i)^k\mathscr{H}_k$.