Abstract
We prove that a sequence (f i ) ∞ i=1 of translates of a fixed f ∈ L p (ℝ) cannot be an unconditional basis of L p (ℝ) for any 1 ≤ p < ∞. In contrast to this, for every 2 < p < ∞, d ∈ ℕ and unbounded sequence (λ n ) n∈ℕ ⊂ ℝd we establish the existence of a function f ∈ L p (ℝd) and sequence (g n *) n∈ℕ ⊂ L p *(ℝd) such that \({({T_{{\lambda _n}}}f,g_n^*)_{n \in {\Bbb N}}}\) forms an unconditional Schauder frame for L p (ℝd). In particular, there exists a Schauder frame of integer translates for L p (ℝ) if (and only if) 2 < p < ∞.
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Research of the first, second, and third author was supported by the National Science Foundation.
Edward Odell (1947–2013). The author passed away during the production of this paper.
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Freeman, D., Odell, E., Schlumprecht, T. et al. Unconditional structures of translates for L p (ℝd). Isr. J. Math. 203, 189–209 (2014). https://doi.org/10.1007/s11856-014-1084-1
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DOI: https://doi.org/10.1007/s11856-014-1084-1