Lower-Order Refinements of Greedy Approximation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909967990128640 |
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| author | Beanland, Kevin Chu, Hung Viet Schlumprecht, Thomas Zsák, András |
| author_facet | Beanland, Kevin Chu, Hung Viet Schlumprecht, Thomas Zsák, András |
| contents | For two countable ordinals $α$ and $β$, a basis of a Banach space $X$ is said to be $(α, β)$-quasi-greedy if it is
1) quasi-greedy,
2) $\mathcal{S}_α$-unconditional but not $\mathcal{S}_{α+1}$-unconditional, and
3) $\mathcal{S}_β$-democratic but not $\mathcal{S}_{β+1}$-democratic.
If $α$ or $β$ is replaced with $\infty$, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a $(0,0)$-quasi-greedy basis, an $(α, \infty)$-quasi-greedy basis, and an $(\infty, α)$-quasi-greedy basis. In this paper, we construct $(α, β)$-quasi-greedy bases for $β\le α+1$ (except the already solved case $α= β= 0$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05533 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower-Order Refinements of Greedy Approximation Beanland, Kevin Chu, Hung Viet Schlumprecht, Thomas Zsák, András Functional Analysis 41A65, 46B15 For two countable ordinals $α$ and $β$, a basis of a Banach space $X$ is said to be $(α, β)$-quasi-greedy if it is 1) quasi-greedy, 2) $\mathcal{S}_α$-unconditional but not $\mathcal{S}_{α+1}$-unconditional, and 3) $\mathcal{S}_β$-democratic but not $\mathcal{S}_{β+1}$-democratic. If $α$ or $β$ is replaced with $\infty$, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a $(0,0)$-quasi-greedy basis, an $(α, \infty)$-quasi-greedy basis, and an $(\infty, α)$-quasi-greedy basis. In this paper, we construct $(α, β)$-quasi-greedy bases for $β\le α+1$ (except the already solved case $α= β= 0$). |
| title | Lower-Order Refinements of Greedy Approximation |
| topic | Functional Analysis 41A65, 46B15 |
| url | https://arxiv.org/abs/2504.05533 |