# Uniform approximation of $L^2$ basis by smooth functions with bounded derivatives of all orders

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 $$\displaystyle\left|\frac{d^k g_i}{dx^k}\right|_\infty
2. For each $$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.

• 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 $f_n(x) = \sum_m \hat{f}(m) e^{2i \pi mx} \hat{\varphi}( m/n)$ – reuns May 4 at 22:59
• Thanks, in that case I'm pretty sure the answer to my question is negative. – ougoah May 5 at 20:52
• 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 – reuns May 5 at 20:58