Lawson's iteration is a classical and effective method for solving the linear (polynomial) minimax approximation in the complex plane. Extension of Lawson's iteration for the rational minimax approximation with both computationally high efficiency and theoretical guarantee is challenging. A recent work [L.-H. Zhang, L. Yang, W. H. Yang and Y.-N. Zhang, A convex dual programming for the rational minimax approximation and Lawson's iteration, 2023, arXiv:2308.06991v1] reveals that Lawson's iteration can be viewed as a method for solving the dual problem of the original rational minimax approximation, and a new type of Lawson's iteration was proposed. Such a dual problem is guaranteed to obtain the original minimax solution under Ruttan's sufficient condition, and numerically, the proposed Lawson's iteration was observed to converge monotonically with respect to the dual objective function. In this paper, we perform theoretical convergence analysis for Lawson's iteration for both the linear and rational minimax approximations. In particular, we show that (i) for the linear minimax approximation, the near-optimal Lawson exponent $\beta$ in Lawson's iteration is $\beta=1$, and (ii) for the rational minimax approximation, the proposed Lawson's iteration converges monotonically with respect to the dual objective function for any sufficiently small $\beta>0$, and the limit approximant fulfills the complementary slackness: any node associated with positive weight either is an interpolation point or has a constant error.

翻译：Lawson迭代是复平面上求解线性（多项式）极小极大逼近问题的经典有效方法。将Lawson迭代拓展至有理极小极大逼近，同时兼顾计算高效性与理论保障，是一项具有挑战性的任务。近期研究工作[张丽环、杨磊、杨卫华、张亚楠，《有理极小极大逼近的凸对偶规划与Lawson迭代》，2023，arXiv:2308.06991v1]揭示了Lawson迭代可视为求解原始有理极小极大逼近对偶问题的一种方法，并提出了一种新型Lawson迭代。在Ruttan充分条件下，该对偶问题能保证获得原始极小极大解，且数值实验表明所提出的Lawson迭代在目标函数上单调收敛。本文对线性与有理极小极大逼近的Lawson迭代进行理论收敛性分析。具体而言，我们证明：(i) 对于线性极小极大逼近，Lawson迭代中最优近似的Lawson指数$\beta$取值为$\beta=1$；(ii) 对于有理极小极大逼近，所提出的Lawson迭代在任意充分小的$\beta>0$下关于对偶目标函数单调收敛，且极限逼近满足互补松弛条件：任何具有正权重的节点要么是插值点，要么具有恒定误差。