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

Let $1\le p\le 2$ and let $\mathrm{\Lambda }=\{{\lambda }_n{\}}_{n\in \mathbb{N}}\subseteq \mathbb{R}$ 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 $p\le q\le p^{\mathrm{\prime }}$, where $p^{\mathrm{\prime }}$ is the conjugate exponent of $p\mathrm{.}$ 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 $M^p(\mathbb{R})\mathrm{.}$ 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 $2<p<\mathrm{\infty }$ will be presented as well.