This paper is about the characterization of the rate of best polynomial approximation on a domain in ${\mathbf{R}}^d$ by multivariate polynomials of degree $n$ via a suitable modulus of continuity or smoothness. We shall be concerned with the situations: approximation on convex polytopes and approximation on (not necessarily convex) smooth algebraic domains. We shall introduce appropriate moduli of smoothness and prove the corresponding Jackson theorem. The usual weak converses are also true.

中文翻译：

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多个变量的多项式逼近

本文讨论了一个域上最佳多项式逼近率的刻画。 ${\mathbf{[R}}^d$ 通过度的多元多项式 $ñ$通过合适的连续性或平滑度 我们将关注以下情况：凸多边形上的逼近和（不一定是凸）光滑代数域上的逼近。我们将引入适当的光滑度模并证明相应的杰克逊定理。通常的弱交谈也是如此。