The $$L_q$$ -weighted dual programming of the linear Chebyshev approximation and an interior-point method

Abstract

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.