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.
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