Existence of Unconditional Frames Formed By System of Translates in Modulation Spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866929711432597504 |
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| author | Yu, Pu-Ting |
| author_facet | Yu, Pu-Ting |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of Unconditional Frames Formed By System of Translates in Modulation Spaces Yu, Pu-Ting Functional Analysis 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. |
| title | Existence of Unconditional Frames Formed By System of Translates in Modulation Spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2502.01047 |