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.org/pdf/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 convergent solution fulfills the complementary slackness: all nodes associated with positive weights achieve the maximum error.

翻译：Lawson迭代是求解复平面内线性（多项式）极小极大逼近问题的经典有效方法。将Lawson迭代扩展至有理数极小极大逼近，并同时保持高计算效率与理论保证颇具挑战性。近期工作[张丽华、杨磊、杨文浩、张亚楠，有理数极小极大逼近与Lawson迭代的凸对偶规划，2023，arxiv.org/pdf/2308.06991v1]揭示了Lawson迭代可视为求解原始有理数极小极大逼近对偶问题的方法，并提出了新型Lawson迭代。在Ruttan充分条件下，该对偶问题可保证获得原始极小极大解，且数值实验表明所提出的Lawson迭代在对偶目标函数上呈单调收敛。本文针对线性与有理数极小极大逼近中的Lawson迭代进行理论收敛性分析。特别地，我们证明了：(i) 对于线性极小极大逼近，Lawson迭代中近优Lawson指数$\beta$的取值为$\beta=1$；(ii) 对于有理数极小极大逼近，当$\beta>0$充分小时，所提出的Lawson迭代在对偶目标函数上单调收敛，且收敛解满足互补松弛条件：所有正权值节点均达到最大误差。