论文标题
在伯恩斯坦类型的定量估计值中
On Bernstein type quantitative estimates for Ornstein non-inequalities
论文作者
论文摘要
对于多indexes的顺序，$ \ {α_i\} _ {i = 1}^{m} $和$β$和$β$和$β$，我们研究不等式\ [\ | d^βf \ | ____ {l_1（l_1（l_1） \ | d^{α_j} f \ | _ {l_1（\ mathbb {t}^d）}，\]，其中$ f $是$ d $ d $ dimensional torus on $ d $ n $的三角学多项式。 Assuming some natural geometric property of the set $\{α_j\}\cup\{β\}$ we show that \[ K_{N}\geq C \left(\ln N\right)^ϕ, \] where $ϕ<1$ depends only on the set $\{α_j\}\cup\{β\}$.
For the sequence of multi-indexes $\{α_i\}_{i=1}^{m}$ and $β$ we study the inequality \[ \|D^β f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{α_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial of degree at most $N$ on $d$-dimensional torus. Assuming some natural geometric property of the set $\{α_j\}\cup\{β\}$ we show that \[ K_{N}\geq C \left(\ln N\right)^ϕ, \] where $ϕ<1$ depends only on the set $\{α_j\}\cup\{β\}$.
