On best uniform approximation of finite sets by linear combinations of real valued functions using linear programming

Abstract

We study the best approximation problem: \[ \displaystyle \min_{\alpha\in \mathbb R^m}\max_{1\leq i\leq n}\left|y_i -\sum_{j=1}^m \alpha_j \Gamma_j ({\bf x}_i) \right|. \] Here: $\Gamma:=\left\{\Gamma_1,...,\Gamma_m\right\}$ is a list of functions where for each $1\leq j\leq m$, $\Gamma_j:\Delta \rightarrow \mathbb R$ with $\Delta$ a set of evaluation points $\left\{{\bf x_1},...,{\bf x_n}\right\}$. $\left\{y_1,...,y_n\right\}$ is a set of real values and $\mathbb R^m:=\left\{(\alpha_1,...,\alpha_m),\, \alpha_j\in \mathbb R,\, 1\leq j\leq m\right\}$.