Given samples of a real or complex-valued function on a set of distinct nodes, the traditional linear Chebyshev approximation is to compute the best minimax approximation on a prescribed linear functional space. Lawson's iteration is a classical and well-known method for that task. However, Lawson's iteration converges linearly and in many cases, the convergence is very slow. In this paper, by the duality theory of linear programming, we first provide an elementary and self-contained proof for the well-known Alternation Theorem in the real case. Also, relying upon the Lagrange duality, we further establish an $L_q$-weighted dual programming for the linear Chebyshev approximation. In this framework, we revisit the convergence of Lawson's iteration, and moreover, propose a Newton type iteration, the interior-point method, to solve the $L_2$-weighted dual programming. Numerical experiments are reported to demonstrate its fast convergence and its capability in finding the reference points that characterize the unique minimax approximation.

翻译：给定实值或复值函数在互异节点集上的采样值，经典线性切比雪夫逼近旨在预设线性函数空间中计算最优极小极大逼近。Lawson迭代是处理此问题的经典方法，但其线性收敛速度在许多情形下极其缓慢。本文首先利用线性规划对偶理论，给出实情形下著名的交替定理（Alternation Theorem）的一个初等且自包含的证明。进一步基于拉格朗日对偶性，为线性切比雪夫逼近建立了$L_q$加权对偶规划。在此框架下，我们重新审视了Lawson迭代的收敛性，并提出牛顿型迭代——内点法，以求解$L_2$加权对偶规划。数值实验展示了该方法快速收敛的特性及其在确定刻画唯一极小极大逼近的参考点方面的能力。