Bernstein polynomials provide a constructive proof for the Weierstrass approximation theorem, which states that every continuous function on a closed bounded interval can be uniformly approximated by polynomials with arbitrary accuracy. Interestingly the proof of this result can be done using elementary probability theory. This way one can even get error bounds for Lipschitz functions. In this note, we present these techniques and show how the method can be extended naturally to other interesting situations. As examples, we obtain in an elementary way results for the Sz\'{a}sz-Mirakjan operator and the Baskakov operator.

翻译：伯恩斯坦多项式为魏尔斯特拉斯逼近定理提供了一种构造性证明，该定理指出：闭区间上的任意连续函数均可被多项式以任意精度一致逼近。有趣的是，该定理的证明可借助初等概率论完成，甚至能为Lipschitz函数导出误差界。本文介绍这些技巧，并展示如何将该方法自然地推广至其他有趣情形。作为范例，我们以初等方式获得了Szász-Mirakjan算子和Baskakov算子的相关结果。