Consider the space $H = L^2(\mathbb{R},e^{-x^2}dx)$ to which belong all polynomials. The set $\{x^n\}_{n = 0,1,2,3,...}$ is basis in this space. However, it is not orthogonal. An orthogonal basis would be the Hermite polynomials, this I know. My problem is that I want to construct a function $f_m(x) \in H$ such that
\begin{equation} \intop_{-\infty}^{+\infty}f_m(x)x^n e^{-x^2}dx \propto \delta_{nm} \mbox{ .} \end{equation}
Any idea how one construct such a function?
