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 any function which is in $L^2([0,T])$ and is cadlag?

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 a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ of piecewise cadlag function which are smooth on each piece.

My question is there a necessary and sufficient condition on the coefficients $\{c_i\}$ of the basis characterizing any function which is in $L^2([0,T])$ and is cadlag?