We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.

翻译：本文研究了凸域上连续函数的最佳一致逼近问题。逼近采用有限函数系统（不一定是切比雪夫系统）的线性组合，并在任意线性约束下进行。通过改进交错点概念和Remez迭代过程，我们提出了一种方法，证明了其在数值问题中的有效性。在有利假设下，证明了线性收敛速率。特别关注复指数系统、高斯函数、稀疏代数和三角多项式。考虑了在信号处理、线性常微分方程、切换动力系统以及Markov-Bernstein型不等式中的应用。