Modifications of the first Remez algorithm

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.