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


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.

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Transactions of the American Mathematical Society