On Bernstein type quantitative estimates for Ornstein non-inequalities
| Autor: | Kazaniecki, Krystian, Wojciechowski, Michał |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | For the sequence of multi-indexes $\{\alpha_i\}_{i=1}^{m}$ and $\beta$ we study the inequality \[ \|D^{\beta} f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{\alpha_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial of degree at most $N$ on $d$-dimensional torus. Assuming some natural geometric property of the set $\{\alpha_j\}\cup\{\beta\}$ we show that \[ K_{N}\geq C \left(\ln N\right)^{\phi}, \] where $\phi<1$ depends only on the set $\{\alpha_j\}\cup\{\beta\}$. Comment: Presentation improved, typos corrected |
| Databáze: | arXiv |
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