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

TL;DR

The paper investigates whether a translate system $\{T_{\lambda_n}g\}_{n\in\mathbb{N}}$ can form unconditional bases or frames in modulation spaces $M^p(\mathbb{R})$. It establishes nonexistence results for $1\le p\le 2$: no unconditional basis for $M^q(\mathbb{R})$ with $p\le q\le p'$, and no unconditional frames for $1<p\le 2$; conversely, unconditional frames from translates can exist for $p>2$. The analysis combines Fourier-invariance of modulation spaces, density/discreteness considerations, and probabilistic and time-frequency techniques (Khintchine inequality, Rademacher systems, Gabor expansions) to delineate when translate systems yield stable, permutation-insensitive representations. These results clarify fundamental limits on unconditional time-frequency expansions in modulation spaces and have implications for the construction of robust signal representations in time-frequency analysis.

Abstract

Let $1\leq p\leq 2$ and let $Λ= \{λ_n\}_{n\in \mathbb{N}} \subseteq \mathbb{R}$ 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 $\{g(x-λ_n)\}_{n\in \mathbb{N}}$ 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 $\{g(x-λ_n)\}_{n\in \mathbb{N}}$ 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.

Theorems & Definitions (38)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof