Representing Systems of Dilations and Translations in Symmetric Function Spaces

Abstract

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 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 1 0 f (t) dt = 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 ).

Sequences of dilations and translations equivalent to the Haar system in $L^p$-spaces

“…It was started as long ago as in 1941 by Men'shov in the case of the trigonometric system [18], and then was continued by Talalyan [25], which has introduced explicitly the notion of representing system (see also the survey [29]). Concerning the study of representing systems of dilations and translations in the L p -spaces see the paper [7] (and references therein) as well the recent work [5], where this problem is considered in a more general setting of rearrangement invariant spaces.…”