Hermite polynomials approximate of a function and its derivatives

Question

Given a differentiable function $f\in \mathcal C^{(n)}(\mathbb R) \cap L^2(\mathbb R,e^{-x^2/2}dx)$ and its Hermite polynomial expansion $f_n=\sum_{i=0}^n a_i \psi_i$. Is it true that $\int_{-\infty}^\infty |f^{(k)}(x)-f_n^{(k)}(x)|^2e^{-\frac{x^2}2}dx\to 0$ where $g^{(k)}$ is the $k$'th derivative of $g$, as $n\to\infty$, $\forall 0\le k\le n$? What is the proof? Is there a general result regarding the convergence of spanning orthogonal polynomial to the derivatives of the original function?

Answer 1

I got it. This is true for Hermite functions. Here is the proof.

For the inner product $\langle\cdot,\cdot\rangle$ by integration by parts $$\langle f', \psi_n\rangle=-\langle f, \psi'_n\rangle,$$ $$\psi_n'=\sqrt\frac n2\psi_{n-1}-\sqrt\frac {n+1}2\psi_{n+1}.$$ \begin{align} f'(x)&=\sum_n \langle f',\psi_n \rangle\psi_n \\ &=\sum_n\bigg(-\sqrt\frac n2\langle f,\psi_{n-1}\rangle\psi_n+\sqrt\frac {n+1}2\langle f,\psi_{n+1}\rangle\psi_n\bigg) \\ &=\sum_n\bigg(-\sqrt\frac {n+1}2\langle f,\psi_n\rangle\psi_{n+1}+\sqrt\frac n2\langle f,\psi_n\rangle\psi_{n-1}\bigg) \\ &= \sum_n\langle f,\psi_n\rangle\psi'_n. \end{align}