Representing Systems of Dilations and Translations in Symmetric Function Spaces

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 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 ∈ X is a representing system in the space X . The main result reads that this holds whenever ∫ 0 1 f ( t ) d t ≠ 0 and f ∈ M ( X ) . Moreover, the condition f ∈ M ( X ) turns out to be sharp in a certain sense. In particular, we prove that a decreasing nonnegative function f , f ≠ 0 , from a Lorentz space Λ φ generates an absolutely representing system of dyadic dilations and translations in Λ φ if and only if f ∈ M ( Λ φ ) .