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Unconditional Schauder frames of translates in \(L_p ( \mathbb{R}^d)\). (English) Zbl 1467.46013

Summary: We show that, for \(1 < p \leq 2\), the space \(L_p ( \mathbb{R}^d)\) does not admit unconditional Schauder frames \(\{f_i, f'_i\}_{i \in \mathbb{N}}\) where \(\{f_i\}\) is a sequence of translates of finitely many functions and \(\{f'_i\}\) is seminormalized. In fact, the only subspaces of \(L_p ( \mathbb{R}^d )\) admitting such Banach frames are those isomorphic to \(\ell_p\). On the other hand, if \(2 < p < +\infty\) and \(\{\lambda_i\}_{i \in \mathbb{N}} \subseteq \mathbb{R}^d\) is an unbounded sequence, there is a subsequence \(\{\lambda_{m_i}\}_{i \in \mathbb{N}}\), a function \(f \in L_p ( \mathbb{R}^d)\), and a seminormalized sequence of bounded functionals \(\{f'_i\}_{i \in \mathbb{N}}\) such that \(\{T_{\lambda_{m_i}} f, f_i^\prime\}_{i \in \mathbb{N}}\) is an unconditional Schauder frame for \(L_p ( \mathbb{R}^d)\).

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
42C15 General harmonic expansions, frames
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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