Existence of Unconditional Frames Formed By System of Translates in Modulation Spaces

Abstract

Let $1\le p\le 2$ and let $\Lambda =\{{\lambda }_n{\}}_{n\in \mathbb{N}}\subseteq \mathbb{R}$ be an arbitrary subset. We prove that for any $g\in M^p(\mathbb{R})$ with $1\le p\le 2$ the system of translates $\{g(x-{\lambda }_n){\}}_{n\in \mathbb{N}}$ is never an unconditional basis for $M^q(\mathbb{R})$ for $p\le q\le p^{\prime }$, where $p^{\prime }$ is the conjugate exponent of $p.$ In particular, $M^1(\mathbb{R})$ does not admit any Schauder basis formed by a system of translates. We also prove that for any $g\in M^p(\mathbb{R})$ with $1<p\le 2$ the system of translates $\{g(x-{\lambda }_n){\}}_{n\in \mathbb{N}}$ is never an unconditional frame for $M^p(\mathbb{R}).$ Several results regarding the existence of unconditional frames formed by a system of translates in $M^1(\mathbb{R})$ as well as in $M^p(\mathbb{R})$ with $2<p<\infty$ will be presented as well.

arXiv comment21 pages. Some citation errors were corrected. We thank Nir Lev for pointing these errors out. Any comment would be greatly appreciated