Unconditional Frames of Translates in $L_p({\mathbb{R}}^d)$

Abstract

We show that, for $1<p\le 2$, the space $L_p({\mathbb{R}}^d)$ does not admit unconditional Schauder frames ${\left\{f_i,f_i^{\prime }\right\}}_{i\in \mathbb{N}}$ where $\left\{f_i\right\}$ is a sequence of translates of finitely many functions and $\left\{f_i^{\prime }\right\}$ 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 ${\left\{{\lambda }_i\right\}}_{i\in \mathbb{N}}\subseteq {\mathbb{R}}^d$ is an unbounded sequence, there is a subsequence ${\left\{{\lambda }_{m_i}\right\}}_{i\in \mathbb{N}}$, a function $f\in L_p({\mathbb{R}}^d)$, and a seminormalized sequence of bounded functionals ${\left\{f_i^{\prime }\right\}}_{i\in \mathbb{N}}$ such that ${\left\{T_{{\lambda }_{m_i}}f,f_i^{\prime }\right\}}_{i\in \mathbb{N}}$ is an unconditional Schauder frame for $L_p({\mathbb{R}}^d)$.

Changes in the new revision: some typos in the bibliography are fixed. Changes in the revised version: The proofs of Remark 2.2, Lemma 3.7 and Proposition 4.2 are slightly improved; some typos are fixed, article format is fixed