Let $\mathcal{H}$ be the space of all functions on the real line $\mathbb{R}$ of the form $p(x) e^{-x^{2} / 2}$, where $p$ is a polynomial with complex coefficients. Make $\mathcal{H}$ into an inner-product space, in the usual way, by defining the inner product to be

$$\langle f, g\rangle=\int_{-\infty}^{\infty} f(t) \overline{g(t)} d t, \quad f, g \in \mathcal{H}$$

You should assume, without proof, that this equation does define an inner product on $\mathcal{H}$. Define the norm by $\|f\|_{2}=\langle f, f\rangle^{1 / 2}$ for $f \in \mathcal{H}$. Now define a sequence of functions $\left(F_{n}\right)_{n \geqslant 0}$ on $\mathbb{R}$ by

$$F_{n}(x)=(-1)^{n} e^{x^{2} / 2} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$$

Prove that $\left(F_{n}\right)$ is an orthogonal sequence in $\mathcal{H}$ and that it spans $\mathcal{H}$.

For every $f \in \mathcal{H}$ define the Fourier transform $\widehat{f}$ of $f$ by

$$\widehat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) e^{-i t x} d x, \quad t \in \mathbb{R}$$

Show that

(a) $\widehat{F}_{n}=(-i)^{n} F_{n}$ for $n=0,1,2, \ldots$;

(b) for all $f \in \mathcal{H}$ and $x \in \mathbb{R}$,

$$\widehat{\widehat{f}}(x)=f(-x)$$

(c) $\|\widehat{f}\|_{2}=\|f\|_{2}$ for all $f \in \mathcal{H}$.