The best uniform rational approximation of the \emph{sign} function on two intervals separated by zero was explicitly solved by E.I. Zolotar\"ev in 1877. This optimization problem is the initial step in the staircase of the so called approximation problems for multiband filters which are of great importance for electrical engineering. We show that known in the literature optimality criterion for this problem may be contradictory since it does not take into account the projective invariance of the problem. We propose a new consistently projective formulation of this problem and give a constructive optimality criterion for it.

中文翻译：

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多带滤波器优化问题的投影视图

EI Zolotar\"ev 在 1877 年明确解决了\emph{sign} 函数在被零隔开的两个区间上的最佳均匀有理近似。这个优化问题是所谓的多带滤波器近似问题阶梯的第一步这对电气工程非常重要。我们表明，在文献中已知的这个问题的最优准则可能是矛盾的，因为它没有考虑到问题的投影不变性。我们提出了这个问题的一个新的一致投影公式，并给出一个建设性的最优标准。