On the cardinality of lower sets and universal discretization
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909576990818304 |
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| author | Dai, F. Prymak, A. Shadrin, A. Temlyakov, V. Tikhonov, S. |
| author_facet | Dai, F. Prymak, A. Shadrin, A. Temlyakov, V. Tikhonov, S. |
| contents | A set $Q$ in $\mathbb{Z}_+^d$ is a lower set if $(k_1,\dots,k_d)\in Q$ implies $(l_1,\dots,l_d)\in Q$ whenever $0\le l_i\le k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $\mathbb{Z}_+^d$. Next we apply these results for universal discretization of the $L_2$-norm of elements from $n$-dimensional subspaces of trigonometric polynomials generated by lower sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_02113 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the cardinality of lower sets and universal discretization Dai, F. Prymak, A. Shadrin, A. Temlyakov, V. Tikhonov, S. Numerical Analysis Primary: 65J05, Secondary: 05A17, 42B05, 65D30, 41A17, 41A63 A set $Q$ in $\mathbb{Z}_+^d$ is a lower set if $(k_1,\dots,k_d)\in Q$ implies $(l_1,\dots,l_d)\in Q$ whenever $0\le l_i\le k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $\mathbb{Z}_+^d$. Next we apply these results for universal discretization of the $L_2$-norm of elements from $n$-dimensional subspaces of trigonometric polynomials generated by lower sets. |
| title | On the cardinality of lower sets and universal discretization |
| topic | Numerical Analysis Primary: 65J05, Secondary: 05A17, 42B05, 65D30, 41A17, 41A63 |
| url | https://arxiv.org/abs/2208.02113 |