Entropy numbers of finite-dimensional Lorentz space embeddings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918128626171904 |
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| author | Prochno, Joscha Sonnleitner, Mathias Vybíral, Jan |
| author_facet | Prochno, Joscha Sonnleitner, Mathias Vybíral, Jan |
| contents | The sequence of entropy numbers quantifies the degree of compactness of a linear operator acting between quasi-Banach spaces. We determine the asymptotic behavior of entropy numbers in the case of natural embeddings between finite-dimensional Lorentz spaces $\ell_{p,q}^n$ in all regimes; our results are sharp up to constants. This generalizes classical results obtained by Schütt (in the case of Banach spaces) and Edmunds and Triebel, Kühn, as well as Guédon and Litvak (in the case of quasi-Banach spaces) for entropy numbers of identities between finte-dimensional Lebesgue sequence spaces $\ell_p^n$. We employ techniques such as interpolation, volume comparison as well as techniques from sparse approximation and combinatorial arguments. Further, we characterize entropy numbers of embeddings between finite-dimensional symmetric quasi-Banach spaces in terms of best $s$-term approximation numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy numbers of finite-dimensional Lorentz space embeddings Prochno, Joscha Sonnleitner, Mathias Vybíral, Jan Functional Analysis Numerical Analysis 47B06, 46B06 (Primary) 41A46, 46B07, 46A16 (Secondary) The sequence of entropy numbers quantifies the degree of compactness of a linear operator acting between quasi-Banach spaces. We determine the asymptotic behavior of entropy numbers in the case of natural embeddings between finite-dimensional Lorentz spaces $\ell_{p,q}^n$ in all regimes; our results are sharp up to constants. This generalizes classical results obtained by Schütt (in the case of Banach spaces) and Edmunds and Triebel, Kühn, as well as Guédon and Litvak (in the case of quasi-Banach spaces) for entropy numbers of identities between finte-dimensional Lebesgue sequence spaces $\ell_p^n$. We employ techniques such as interpolation, volume comparison as well as techniques from sparse approximation and combinatorial arguments. Further, we characterize entropy numbers of embeddings between finite-dimensional symmetric quasi-Banach spaces in terms of best $s$-term approximation numbers. |
| title | Entropy numbers of finite-dimensional Lorentz space embeddings |
| topic | Functional Analysis Numerical Analysis 47B06, 46B06 (Primary) 41A46, 46B07, 46A16 (Secondary) |
| url | https://arxiv.org/abs/2404.06058 |