Abstract
We show that, for 1 < p ≤ 2, the space Lp(ℝd) does not admit unconditional Schauder frames {fi, f′i}i∈ℕ where {fi} is a sequence of translates of finitely many functions and {f′i} is seminormalized. In fact, the only subspaces of Lp(ℝd) admitting such Banach frames are those isomorphic to ℓp. On the other hand, if 2 < p < +∞ and {λi}i∈ℕ ⊆ ℝd is an unbounded sequence, there is a subsequence {λmi}i∈ℕ, a function f ∈ Lp(ℝd), and a seminormalized sequence of bounded functionals {λ′i}i∈ℕ such that \({\left\{{{T_{{\lambda_{{m_i}}}}}f,f_i^\prime} \right\}_{i \in ℕ}}\) is an unconditional Schauder frame for Lp(ℝd).
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Ackowledgements
We want to thank Daniel Galicer for useful comments, particularly for the one that motivated our Proposition 4.2. We also want to thank the anonymous referee for carefully reading the manuscript and for helpful and detailed comments and suggestions.
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This work was partially supported by CONICET-PIP 11220130100329CO, ANPCyT PICT 2015–2299.
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Berasategui, M., Carando, D. Unconditional Schauder frames of translates in Lp(ℝd). Isr. J. Math. 238, 687–713 (2020). https://doi.org/10.1007/s11856-020-2041-9
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DOI: https://doi.org/10.1007/s11856-020-2041-9