Systems formed by translates of one element in L p ( R ) L_{p}(mathbb R)
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Let 1 p < , f Lp(R), and R. We consider the closed subspace of Lp(R), Xp(f, ), generated by the set of translations f() of f by . If p = 1 and {f(): } is a bounded minimal system in L1(R), we prove that X1(f, ) embeds almost isometrically into 1. If {f(): } is an unconditional basic sequence in Lp(R), then {f(): } is equivalent to the unit vector basis of p for 1 p 2 and Xp(f, ) embeds into p if 2 < p 4. If p > 4, there exists f Lp(R) and Z so that {f(): } is unconditional basic and Lp(R) embeds isomorphically into Xp(f, ). 2011 American Mathematical Society.