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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:

  1. For each integer $k\geq 0$, there exists a constant $C_{\epsilon,k}$ such that for all $i$, we have $$\displaystyle\left|\frac{d^k g_i}{dx^k}\right|_\infty<C_{\epsilon,k}.$$
  2. 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.

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  • $\begingroup$ The best smooth approximation to $f \in L^2$ is $f_n= f \ast n \varphi(n.)$ (convolution) for some fixed $\varphi \in C^\infty_c,\int \varphi=1$. Then $\|f^{(k)}\|_\infty \le \|f\|_{L^2} n^k\| \varphi^{(k)}\|_{L^2}$. In term of Fourier series it becomes $f_n(x) = \sum_m \hat{f}(m) e^{2i \pi mx} \hat{\varphi}( m/n)$ $\endgroup$ – reuns May 4 at 22:59
  • $\begingroup$ Thanks, in that case I'm pretty sure the answer to my question is negative. $\endgroup$ – ougoah May 5 at 20:52
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    $\begingroup$ For $\|f\|_{L^2}$, $\|f_n^{(k)}\|_\infty$ only depends on $k,n$ not on the particular $f$, what depends on $f$ is $\|f-f_n\|_{L^2}$. With $f(x) = e^{2i \pi m x}$ the larger is $m$ the larger you need $n$ to approximate it so the larger will be the derivative $\endgroup$ – reuns May 5 at 20:58

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