Restriction on coefficients in basis of smooth functions

Question

If $f \in L^2([0,T])$ then it can be written as $$ f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t), $$ for some sequence $\{c_i\}$ of real numbers and some Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ of smooth functions.

My question is there a necessary and sufficient condition on the coefficients $\{c_i\}$ of the basis characterizing $C^1$ functions?

Answer 1

I think that will depend from your choice of basis for example for fourier basis (we are in $L^2[0,T]$) we have :

If $f\in \mathcal{C}^k$ then $$ c_n=o\left(\frac{1}{n^k}\right) $$

but

If $$ c_n=O\left(\frac{1}{n^{k+2}}\right) $$ then $$ f\in \mathcal{C}^k $$