Let $\mathcal{F}=\{f_i\}_{i\in\mathbb{N}}$ be an orthonormal Hilbert basis of $L^2[0,1]$.
I am wondering whether it is possible to approximate the $f_i$ uniformly across $i$ in the $L^2$-norm by families of uniformly bounded functions whose derivatives of every order are uniformly bounded.
More precisely,
Question: for every $\epsilon>0$, does there exist a family of smooth functions $\{g_i\}_{i\in\mathbb{N}}$ such that:
- For each integer $k\geq 0$, there exists a constant $C_{\epsilon,k}$ such that for all $i$, we have $$\displaystyle\left|\frac{d^k g_i}{dx^k}\right|_\infty<C_{\epsilon,k}.$$
- For each $i\in\mathbb{N}$, $$||f_i-g_i||_{L^2}<\epsilon?$$
Here the notation $\left|\,\cdot\,\right|_\infty$ means the sup norm of continuous functions on $[0,1]$.
Thoughts: I'm thinking of the case when $\mathcal{F}$ is a $1$-periodic basis of $L^2[0,1]$ consisting of complex exponentials, but because the derivatives of these functions go to $\infty$, I'm not sure how to approximate them using functions $g_i$ of the type given in the question, although I feel this should be possible.