We consider discrete best approximation problems formulated and solved in the framework of tropical algebra that deals with semirings and semifields with idempotent addition. Given a set of samples each consisting of input and output of an unknown function defined on an idempotent semifield, the problems are to find a best approximation of the function by tropical Puiseux polynomial and rational functions. A new solution approach is proposed which involves the reduction of the problem of polynomial approximation to best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation end evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form by using an iterative alternating algorithm. To illustrate the technique developed, we solve example approximation problems in terms of a real semifield where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which correspond to the best Chebyshev approximation by piecewise linear functions.

翻译：我们考虑在热带代数框架下（该框架处理具有幂等加法的半环与半域）表述并求解的离散最佳逼近问题。给定一组由未知函数（定义在幂等半域上）的输入与输出构成的样本，问题旨在通过热带普伊瑟多项式函数与有理函数寻求该函数的最佳逼近。我们提出一种新解法：将多项式逼近问题归约为对一侧含未知向量的热带线性向量方程（单侧方程）的最佳逼近解。我们推导出该单侧方程的解析形式最佳逼近解，并直接以解析形式评估其固有逼近误差。进一步地，我们将有理逼近问题归约为两侧均含未知向量的方程（双侧方程）的最佳逼近解。通过迭代交替算法以数值形式获得双侧方程的最佳逼近解。为阐明该技术，我们以加法定义为最大值、乘法定义为算术加法的实半域（极大-加代数）为背景求解示例逼近问题，该情形对应分段线性函数的最佳切比雪夫逼近。