A convergence analysis of Lawson's iteration for computing polynomial and rational minimax approximations

Lawson's iteration is a classical and effective method for solving the linear (polynomial) minimax approximation problem in the complex plane. Extension of Lawson's iteration for the rational minimax approximation problem 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 problem for the rational minimax approximation and Lawson's iteration, Math. Comp., 94(2025), 2457-2494.] reveals that Lawson's iteration can be viewed as a method for solving the dual problem of the original rational minimax approximation problem, and a new type of Lawson's iteration, namely, d-Lawson, was proposed, which reduces to the classical Lawson's iteration for the linear minimax approximation problem. For the rational case, such a dual problem is guaranteed to obtain the original minimax solution under Ruttan's sufficient condition, and numerically, d-Lawson was observed to converge monotonically with respect to the dual objective function. In this paper, we present a theoretical convergence analysis of d-Lawson for both the linear and rational minimax approximation problems. In particular, we show that (i) for the linear minimax approximation problem, $\beta=1$ is a near-optimal Lawson exponent in Lawson's iteration, and (ii) for the rational minimax approximation problem, under certain conditions, d-Lawson converges monotonically with respect to the dual objective function for any sufficiently small $\beta>0$, and the limiting approximant satisfies the complementary slackness condition: any node associated with positive weight either is an interpolation point or has a constant error.