Gabor unconditional bases and frames in $L^p(\mathbb{R})$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918509066321920 |
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| author | Lev, Nir Tselishchev, Anton |
| author_facet | Lev, Nir Tselishchev, Anton |
| contents | We consider the following problem: given a set $Λ\subset \mathbb{R} \times \mathbb{R}$ and $p \neq 2$, does there exist a function $g \in L^p(\mathbb{R})$ such that the Gabor system $\{g(x-t) e^{2 πisx}\}$, $(t,s) \in Λ$, consisting of time-frequency shifts of $g$, forms an unconditional basis or unconditional Schauder frame in the space $L^p(\mathbb{R})$? We completely resolve this question for $p>2$; in particular, we characterize the sets $Λ$ such that an unconditional Schauder frame of this form exists. We also prove a Balian-Low type result, showing that the window function $g$ cannot enjoy mild continuity and decay conditions. For $1<p<2$, we prove that a Gabor system cannot form an unconditional basis or unconditional Schauder frame in $L^p(\mathbb{R})$ if the set $Λ$ satisfies a natural separation condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gabor unconditional bases and frames in $L^p(\mathbb{R})$ Lev, Nir Tselishchev, Anton Classical Analysis and ODEs Functional Analysis 42C15, 46B15, 46E30 We consider the following problem: given a set $Λ\subset \mathbb{R} \times \mathbb{R}$ and $p \neq 2$, does there exist a function $g \in L^p(\mathbb{R})$ such that the Gabor system $\{g(x-t) e^{2 πisx}\}$, $(t,s) \in Λ$, consisting of time-frequency shifts of $g$, forms an unconditional basis or unconditional Schauder frame in the space $L^p(\mathbb{R})$? We completely resolve this question for $p>2$; in particular, we characterize the sets $Λ$ such that an unconditional Schauder frame of this form exists. We also prove a Balian-Low type result, showing that the window function $g$ cannot enjoy mild continuity and decay conditions. For $1<p<2$, we prove that a Gabor system cannot form an unconditional basis or unconditional Schauder frame in $L^p(\mathbb{R})$ if the set $Λ$ satisfies a natural separation condition. |
| title | Gabor unconditional bases and frames in $L^p(\mathbb{R})$ |
| topic | Classical Analysis and ODEs Functional Analysis 42C15, 46B15, 46E30 |
| url | https://arxiv.org/abs/2605.17970 |