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：本文研究凸域上连续函数的最佳一致逼近问题。该逼近是在任意线性约束条件下，通过有限函数系（不一定是切比雪夫系）的线性组合实现的。通过修改交错点概念和Remez迭代过程，我们提出了一种在数值问题中展现高效性的方法。在某些有利假设下，证明了该方法的线性收敛速率。研究特别关注复指数系、高斯函数系、缺项代数与三角多项式系。探讨了该方法在信号处理、线性常微分方程、切换动力系统以及马尔可夫-伯恩斯坦型不等式中的应用。