LAUSR

Let X be an arbitrary separable symmetric function space on [0, 1]. By using a combination of the frame approach and the notion of the multiplicator space $$\mathscr {M}(X)$$ 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 $$f\in X$$ f ∈ X is a representing system in the space X . The main result reads that this holds whenever $$\int _0^1 f(t)\,dt\ne 0$$ ∫ 0 1 f ( t ) d t ≠ 0 and $$f\in \mathscr {M}(X).$$ f ∈ M ( X ) . Moreover, the condition $$f\in \mathscr {M}(X)$$ f ∈ M ( X ) turns out to be sharp in a certain sense. In particular, we prove that a decreasing nonnegative function f ,  $$f\ne 0,$$ f ≠ 0 , from a Lorentz space $$\varLambda _{\varphi }$$ Λ φ generates an absolutely representing system of dyadic dilations and translations in $$\varLambda _{\varphi }$$ Λ φ if and only if $$f\in \mathscr {M}(\varLambda _{\varphi }).$$ f ∈ M ( Λ φ ) .