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

TL;DR

This paper investigates the limitations of using systems of translates as unconditional bases or frames in modulation spaces, showing non-existence results for certain ranges of p and q, and exploring conditions for unconditional frames.

Contribution

It establishes new non-existence results for unconditional bases and frames formed by translates in modulation spaces, extending understanding of their structure and limitations.

Findings

No unconditional basis of translates in M^p for 1 ≤ p ≤ 2.

No unconditional frame of translates in M^p for 1 < p ≤ 2.

Results on existence of unconditional frames in M^1 and M^p for p>2.

Abstract

Let $1\leq p\leq 2$ 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 $1\leq p\leq 2$ the system of translates $\{g(x-\lambda_n)\}_{n\in \mathbb{N}}$ is never an unconditional basis for $M^q(\mathbb{R})$ for $p\leq q\leq p'$, where $p'$ 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 $1< p\leq 2$ 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.