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 `1. 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 `p for 1 ≤ p ≤ 2 and Xp(f,Λ) embeds into `p 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).