局部各向异性多项式L_p-逼近，0<p<1的Whitney定理
Whitney's theorem for local anisotropic polynomial L_p-approximation, 0<p<1
摘要：
Dinh Dng和T. Ullrich已经证明了一个多元惠特尼定理的局部各向异性多项式近似的$L_p(Q)$为$1 \le p \le \infty$，其中$Q$是一个$d$ -平行六面体在$\RR^d$与平行的坐标轴。他们认为错误的最佳逼近的功能$f$代数多项式的固定程度最$r_i - 1$在变量$x_i,\ i=1,...,d$。当$Q$的大小变为0时的逼近误差的收敛速率由所谓的全混合光滑模表征。这些作者所使用的证明方法不适用于$0 <p<1$的情况。本文对$0< p \le \infty$用另一种方法证明了这个定理。
英文摘要：
Dinh Dũng and T. Ullrich have proven a multivariate Whitney's theorem for the local anisotropic polynomial approximation in $L_p(Q)$ for $1 \le p \le \infty$, where $Q$ is a $d$-parallelepiped in $\RR^d$ with sides parallel to the coordinate axes. They considered the error of best approximation of a function $f$ by algebraic polynomials of fixed degree at most $r_i - 1$ in variable $x_i,\ i=1,...,d$. The convergence rate of the approximation error when the size of $Q$ going to 0 is characterized by a so-called total mixed modulus of smoothness. The method of proof used by these authors is not suitable to the case $0 <p<1$. In the present paper, by a different method we proved this theorem for $0< p \le \infty$.
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