限制偏导数的函数的多项式逼近

<正> 1.引言在一元函数中用一个与它单调性相同的多项式来逼近的问题已经有许多数学工作者作了精辟的论述1——16]。作者在17]中研究了二维偏单调函数的共单调多项式逼近。本文研究了限制偏导数的函数的多项式逼近,得出了与偏共单调逼近有关的一些结论。

POLYNOMIAL APPROXIMATION TO FUNCTION WITH RESTRICTED PARTIAL DERIVATIVES

This paper studies the problem of polynomial approximation to function with restricted partial derivatives. It is shown the following theorem. Theorem 4. Let Assume that for all X in D we have for j=1, …, r, Then for n=0, 1, 2, … there is a polynomial (X) of best approximation to f on D. For n sufficiently large we have for X= (x_1, …, x_) ∈D, where J_1+…+j_k=1_j and 2j≤x (i=1, …, k). It is also shown a simple theorem concerning partial comonotone approximation. Theorem 7. Let f(X) have two continuous partial derivatives on D and assume that for all i=1, …, k in D we have(f(X))/(x_)≥δ>0 and assume f(X) is eparable. Then for n sufficiently large there is a polynomial P_(X) ∈H.(X) for all i=1,…, k we have (f(X))/(x_)>0, for X∈D.