There are no unconditional Schauder frames of translates in $L^p(\mathbb{R})$, $1 \le p \le 2$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913641740107776 |
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| author | Lev, Nir Tselishchev, Anton |
| author_facet | Lev, Nir Tselishchev, Anton |
| contents | It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space $L^p(\mathbb{R})$ for any $1 \le p < \infty$. To the contrary, there do exist unconditional Schauder frames of translates in $L^p(\mathbb{R})$ for every $p>2$. The existence of such a system for $1 < p \leq 2$, however, has remained an open problem. In this paper the problem is solved in the negative: we prove that none of the spaces $L^p(\mathbb{R})$, $1 \le p \le 2$, admits an unconditional Schauder frame of translates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_01757 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | There are no unconditional Schauder frames of translates in $L^p(\mathbb{R})$, $1 \le p \le 2$ Lev, Nir Tselishchev, Anton Classical Analysis and ODEs Functional Analysis 46B15, 46E30 It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space $L^p(\mathbb{R})$ for any $1 \le p < \infty$. To the contrary, there do exist unconditional Schauder frames of translates in $L^p(\mathbb{R})$ for every $p>2$. The existence of such a system for $1 < p \leq 2$, however, has remained an open problem. In this paper the problem is solved in the negative: we prove that none of the spaces $L^p(\mathbb{R})$, $1 \le p \le 2$, admits an unconditional Schauder frame of translates. |
| title | There are no unconditional Schauder frames of translates in $L^p(\mathbb{R})$, $1 \le p \le 2$ |
| topic | Classical Analysis and ODEs Functional Analysis 46B15, 46E30 |
| url | https://arxiv.org/abs/2312.01757 |