  Let $X$ be an arbitrary separable symmetric space on $[0,1]$. By using a
combination of the frame approach and the notion of the multiplicator space
$\mathscr{M}(X)$ of $X$ with respect to the tensor product, we investigate the
problem when the sequence of dyadic dilations and translations of a function
$f\in X$ is a representing system in the space $X$. The main result reads that
this holds whenever $\int_0^1 f(t)\,dt\ne 0$ and $f\in \mathscr{M}(X)$.
Moreover, the condition $f\in\mathscr{M}(X)$ turns out to be sharp in a certain
sense. In particular, we prove that a decreasing nonnegative function $f$,
$f\ne 0$, from a Lorentz space $\varLambda_{\varphi}$ generates an absolutely
representing system of dyadic dilations and translations in
$\varLambda_{\varphi}$ if and only if $f\in\mathscr{M}(\varLambda_{\varphi})$.
