Representing systems of dilations and translations in symmetric spaces

2019/03/17 by Astashkin, Sergey V., Terekhin, Pavel A.
#42C15 #46B15 #46B70 #46E30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1903.07094

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 \mathscrM(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 ∫₀¹ f(t) dt≠ 0 and f∈ \mathscrM(X). Moreover, the condition f∈\mathscrM(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 \varLambda_φ generates an absolutely representing system of dyadic dilations and translations in \varLambda_φ if and only if f∈\mathscrM(\varLambda_φ).

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