proof for a basis in $L^2$ I know, correct me if I am wrong, that the functions $H_n(x)\exp(-x^2/2)$ form a complete basis in $L^2(\mathbb{R},dx)$, where $H_n(x)$ is the $n$th Hermite polynomial. This must be true also for $x^n\exp(-x^2/2)$ with $n\in\mathbb{N}_0$. Does someone know a proof of the latter or can give me a reference? Since I am not a mathematician, I will really appreciate it if the proof contains all the details that might puzzle a non-mathematician.
I also have the problem to choose functions $f_k$ such that
\begin{equation}
\int_{-\infty}^{+\infty}f_k(x)x^n\exp(-x^2)dx = \delta_{kn} \mbox{.}
\end{equation}
Does anyone know a method to construct the $f_k$s?

Wikipedia has the following  proof of completeness. Suppose $f\in L^2(\mathbb R,\exp(-x^2))$ is orthogonal to all $H_n$, that is $$\int_{\mathbb R} f(x)x^n \exp(-x^2)\,dx=0\quad \text{ for all }\ n\tag1$$ Consider the entire function
$$
F(z)= \int_{\mathbb R} f(x)\exp(zx-x^2)\,dx \tag2
$$
where the integral converges because both $f$ and $e^{zx}$ are in $L^2(\mathbb R,\exp(-x^2))$.  Calculating the $n$th derivative $F^{(n)}(0)$, we obtain
$$
F^{(n)}(0)= \int_{\mathbb R} f(x)x^n\exp( -x^2)\,dx =0 \tag3 
$$
Thus $F\equiv 0$. In particular, $F(iy)\equiv 0$, which says precisely that the Fourier transform of $f(x)\exp(-x^2)$ is zero. By the Fourier inversion formula, $f(x)\exp(-x^2)\equiv 0$, completing the proof.
