Unconditional Schauder frames of exponentials and of uniformly bounded functions in $L^p$ spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911143676608512 |
|---|---|
| author | Lev, Nir Tselishchev, Anton |
| author_facet | Lev, Nir Tselishchev, Anton |
| contents | It is known that there is no unconditional basis of exponentials in the space $L^p(Ω)$, $p \ne 2$, for any set $Ω\subset \mathbb{R}^d$ of finite measure. This is a consequence of a more general result due to Gaposhkin, who proved that the space $L^p(Ω)$ does not admit a seminormalized unconditional basis consisting of uniformly bounded functions. We show that the latter result fails if the word "basis" is replaced with "Schauder frame". On the other hand we prove that if $Ω$ has nonempty interior then there are no unconditional Schauder frames of exponentials in the space $L^p(Ω)$, $p \ne 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02782 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unconditional Schauder frames of exponentials and of uniformly bounded functions in $L^p$ spaces Lev, Nir Tselishchev, Anton Classical Analysis and ODEs Functional Analysis 42A10, 46B15, 46E30 It is known that there is no unconditional basis of exponentials in the space $L^p(Ω)$, $p \ne 2$, for any set $Ω\subset \mathbb{R}^d$ of finite measure. This is a consequence of a more general result due to Gaposhkin, who proved that the space $L^p(Ω)$ does not admit a seminormalized unconditional basis consisting of uniformly bounded functions. We show that the latter result fails if the word "basis" is replaced with "Schauder frame". On the other hand we prove that if $Ω$ has nonempty interior then there are no unconditional Schauder frames of exponentials in the space $L^p(Ω)$, $p \ne 2$. |
| title | Unconditional Schauder frames of exponentials and of uniformly bounded functions in $L^p$ spaces |
| topic | Classical Analysis and ODEs Functional Analysis 42A10, 46B15, 46E30 |
| url | https://arxiv.org/abs/2505.02782 |