Unconditional Schauder frames of translates in Lp(ℝd)

We show that, for 1 < p ≤ 2, the space L p (ℝ d ) does not admit unconditional Schauder frames { f i , f′ i } i ∈ℕ where { f i } is a sequence of translates of finitely many functions and { f′ i } is seminormalized. In fact, the only subspaces of L p (ℝ d ) admitting such Banach frames are those...

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Bibliographic Details
Published in:Israel journal of mathematics 2020-07, Vol.238 (2), p.687-713
Main Authors: Berasategui, Miguel, Carando, Daniel
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Frames
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Subspaces
Theoretical
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Summary:We show that, for 1 < p ≤ 2, the space L p (ℝ d ) does not admit unconditional Schauder frames { f i , f′ i } i ∈ℕ where { f i } is a sequence of translates of finitely many functions and { f′ i } is seminormalized. In fact, the only subspaces of L p (ℝ d ) admitting such Banach frames are those isomorphic to ℓ p . On the other hand, if 2 < p < +∞ and {λ i } i ∈ℕ ⊆ ℝ d is an unbounded sequence, there is a subsequence {λ m i } i ∈ℕ , a function f ∈ L p (ℝ d ), and a seminormalized sequence of bounded functionals {λ′ i } i ∈ℕ such that { T λ m i f , f i ′ } i ∈ ℕ is an unconditional Schauder frame for L p (ℝ d ).
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-020-2041-9