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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.16644 |
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| _version_ | 1866915200822673408 |
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| author | Kesseböhmer, Marc Niemann, Aljoscha |
| author_facet | Kesseböhmer, Marc Niemann, Aljoscha |
| contents | In this note, we present an abstract approach to study asymptotic orders for adaptive approximations with respect to a monotone set function $\mathfrak{J}$ defined on dyadic cubes. We determine the exact upper order in terms of the critical value of the corresponding $\mathfrak{J}$-partition function, and we are able to provide upper and lower bounds in term of fractal-geometric quantities. With properly chosen $\mathfrak{J}$, our new approach has applications in many different areas of mathematics, including the spectral theory of Krein-Feller operators, quantization dimensions of compactly supported probability measures, and the exact asymptotic order for Kolmogorov, Gelfand and linear widths for Sobolev embeddings into $L_μ^p$-spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16644 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exact asymptotic order for generalised adaptive approximations Kesseböhmer, Marc Niemann, Aljoscha Optimization and Control Information Theory Functional Analysis Probability primary: 68W25, 41A25 secondary: 28A80, 65D15 In this note, we present an abstract approach to study asymptotic orders for adaptive approximations with respect to a monotone set function $\mathfrak{J}$ defined on dyadic cubes. We determine the exact upper order in terms of the critical value of the corresponding $\mathfrak{J}$-partition function, and we are able to provide upper and lower bounds in term of fractal-geometric quantities. With properly chosen $\mathfrak{J}$, our new approach has applications in many different areas of mathematics, including the spectral theory of Krein-Feller operators, quantization dimensions of compactly supported probability measures, and the exact asymptotic order for Kolmogorov, Gelfand and linear widths for Sobolev embeddings into $L_μ^p$-spaces. |
| title | Exact asymptotic order for generalised adaptive approximations |
| topic | Optimization and Control Information Theory Functional Analysis Probability primary: 68W25, 41A25 secondary: 28A80, 65D15 |
| url | https://arxiv.org/abs/2312.16644 |