论文标题
$ c^2 $ domains上的多项式近似
Polynomial approximation on $C^2$-domains
论文作者
论文摘要
我们介绍了适当的平滑度计算模量，以表征在连接和紧凑的$ c^2 $ - domain $ω\ subset \ subset \ mathbb {r}^d $上通过多元多项式上的最佳近似速率。这种新的平滑度模量是通过沿坐标轴的方向以及从边界的许多切向方向来定义的。使用此模量，我们证明了直接的杰克逊不等式和相应的逆逆差值（在$ l_p（ω）$中）的相应反向。杰克逊不等式的全部范围为$ 0 <p \ leq \ infty $，而其证明依赖于最近建立的惠特尼类型估计，而常数仅取决于某些参数；在高度局部的多项式分区上，在$ c^2 $域中具有独立利益。以$ 1 \ leq p \ leq \ infty $的价格建立了逆不平等，其证明依赖于最近证明的与$ω$边界上的切向衍生产品相关的伯恩斯坦类型不等式。这样的不平等还使我们能够建立伊万诺夫（Ivanov）在一般紧凑型$ c^2 $ domains上平均平滑度的平均平均模量的逆定理。
We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $Ω\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential directions from the boundary. With this modulus, we prove both the direct Jackson inequality and the corresponding inverse for the best polynomial approximation in $L_p(Ω)$. The Jackson inequality is established for the full range of $0<p\leq \infty$, while its proof relies on a recently established Whitney type estimates with constants depending only on certain parameters; and on a highly localized polynomial partitions of unity on a $C^2$-domain which is of independent interest. The inverse inequality is established for $1\leq p\leq \infty$, and its proof relies on a recently proved Bernstein type inequality associated with the tangential derivatives on the boundary of $Ω$. Such an inequality also allows us to establish the inverse theorem for Ivanov's average moduli of smoothness on general compact $C^2$-domains.
