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 $\Omega\subset \mathbb{R}^d$. 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 $L_p(\Omega)$. 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 $\Omega$. Such an inequality also allows us to establish the inverse theorem for Ivanov's average moduli of smoothness on general compact $C^2$-domains.

翻译：我们引入适当可计算的光滑模来刻画在连通紧致$C^2$区域$\Omega\subset \mathbb{R}^d$上多元多项式最佳逼近的速率。该新型光滑模通过沿坐标轴方向以及沿边界若干切向方向的有限差分类定义。利用此模，我们证明了$L_p(\Omega)$中最佳多项式逼近的直接Jackson不等式及相应的逆定理。Jackson不等式对$0<p\leq \infty$全范围成立，其证明依赖于近期建立的仅与特定参数相关的Whitney型估计，以及$C^2$区域上具有独立兴趣的高度局部化多项式单位分解。逆定理对$1\leq p\leq \infty$成立，其证明依赖于近期证明的与$\Omega$边界上切向导数相关的Bernstein型不等式。该不等式还使我们能够在一般紧致$C^2$区域上建立Ivanov平均光滑模的逆定理。