Abstract
This paper proves the convergence of several modifications of the first Remez algorithm for the solution of linear and nonlinear Chebyshev approximation problems on compact $B \subset \mathbb{R}^s $. While the first Remez algorithm in its original form requires the determination of the global maximum of the error function on all of B in each iteration, the algorithms given here are based on its being sufficient to compute the maximum of the kth error function on a grid $B_{k + 1} $, where $\{ {B_k } \}_{k \geqq 0} $ is a prescribed sequence of finite-point sets in B with density tending to zero. Interpreted differently, some results on the discretization of Chebyshev approximation problems, which do not use full grids $B_k $ in B but only small subsets of them, are provided. The paper concludes with some numerical examples for the solution of linear multivariate problems.
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