We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical analysis, complexity theory, quantum algorithms, etc. The one, most relevant for us, is the amplification of approximation method: it allows to construct approximations of higher degree $M$ and better accuracy from the approximations of degree $m$.

翻译：我们针对不相交线段并集上的局部常数函数，构造了显式且易于实现的高精度多项式逼近。该问题在数值分析、复杂性理论、量子算法等多个领域具有重要应用。其中与我们最相关的一个应用是逼近放大方法：该方法允许从次数为$m$的逼近出发，构造次数为$M$且精度更高的逼近。