Given samples of a real or complex-valued function on a set of distinct nodes, the traditional linear Chebyshev approximation is to compute the minimax approximation on a prescribed linear functional space. Lawson’s iteration is a classical and well-known method for the task. However, Lawson’s iteration converges only linearly and in many cases, the convergence is very slow. In this paper, relying upon the Lagrange duality, we establish an \(L_q\)-weighted dual programming for the discrete linear Chebyshev approximation. In this framework of dual problem, we revisit the convergence of Lawson’s iteration and provide a new and self-contained proof for the well-known Alternation Theorem in the real case; moreover, we 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.

给定一组不同节点上的实值或复值函数的样本，传统的线性切比雪夫近似是在规定的线性函数空间上计算极小极大近似。劳森迭代是该任务的一种经典且众所周知的方法。然而，Lawson 迭代仅线性收敛，并且在许多情况下收敛非常慢。在本文中，依靠拉格朗日对偶性，我们建立了离散线性切比雪夫近似的\(L_q\)加权对偶规划。在这个对偶问题的框架中，我们重新审视了劳森迭代的收敛性，并为著名的交替定理在实际情况中提供了一个新的、独立的证明；此外，我们提出了牛顿型迭代，即内点法，来解决\(L_2\)加权对偶规划。据报道，数值实验证明了其快速收敛性及其寻找表征独特极小极大近似特征的参考点的能力。