I'm confused about Orthonormal basis for $L^2(\mathbb{R})$ by Hermite Polynomials of the answer. ( The satement $ A_{n,m(n)} \rightarrow -\int f(x)^2 e^{-x^2} dx$ )
The following hold?
Let $f \in L^2(\mathbb{R})$ satisfying $ \int_{-\infty}^{\infty} x^nf(x) e^{-x^2}dx=0 \ \forall n \in \mathbb{Z}_{\ge 0}$, and
$(p_n(x))$ be polynomial sequence. Then
$$\lim_{n \to \infty}\int_{-n}^{n}f(x) p_n (x) e^{-x^2}dx =0$$
Any advise would be appreciated.