#### Title

Systems formed by translates of one element in Lp(R)

#### Abstract

Let 1 ≤ p < ∞, f ∈ Lp(R), and Λ ⊆ R. We consider the closed subspace of Lp(R), Xp(f, Λ), generated by the set of translations f(λ) of f by λ ∈ Λ. If p = 1 and {f(λ): λ ∈ Λ} is a bounded minimal system in L1(R), we prove that X1(f, Λ) embeds almost isometrically into ℓ1. If {f(λ): λ ∈ Λ} is an unconditional basic sequence in Lp(R), then {f(λ): λ ∈ Λ} is equivalent to the unit vector basis of ℓp for 1 ≤ p ≤ 2 and Xp(f, Λ) embeds into ℓp if 2 < p ≤ 4. If p > 4, there exists f ∈ Lp(R) and Λ ⊆ Z so that {f(λ): λ ∈ Λ} is unconditional basic and Lp(R) embeds isomorphically into Xp(f, Λ). © 2011 American Mathematical Society.

#### Publication Title

Transactions of the American Mathematical Society

#### Recommended Citation

Odell, E., Sari, B., Schlumprecht, T., & Zheng, B.
(2011). Systems formed by translates of one element in Lp(R).* Transactions of the American Mathematical Society**, 363* (12), 6505-6529.
https://doi.org/10.1090/S0002-9947-2011-05305-5