中文摘要：

该文证明了C[0，1]空间中的函数及其导数可以用Bernstein算子的线性组合同时逼近，得到逼近的正定理与逆定理．同时，也证明了Bernstein算子导数与函数光滑性之间的一个等价关系．该文所获结果沟通了Bernstein算子同时逼近的整体结果与经典的点态结果之间的关系．

英文摘要：

In this paper, we show that the functions in space C[0, 1] and their derivatives can be simultaneously approximated by the combinations of the Bernstein operators. Both direct and inverse theorems are proved. An equivalence relation between the derivatives of the Bernstein operators and the smoothness of function is obtained as well. These results bridge the gap between the classical pointwise conclusions and the global conclusions for simultaneous approximation by the Bernstein operators.