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

Abstract

We study the best approximation problem: \[ \displaystyle \min_{α\in \mathbb R^m}\max_{1\leq i\leq n}\left|y_i -\sum_{j=1}^m α_j Γ_j ({\bf x}_i) \right|. \] Here: $\Gamma :=\left\{{\Gamma }_1,...,{\Gamma }_m\right\}$ is a list of functions where for each $1\le j\le m$, ${\Gamma }_j:\Delta \to \mathbb{R}$ with $\Delta$ a set of evaluation points $\left\{{\mathbf{x}}_1,...,{\mathbf{x}}_{\mathbf{n}}\right\}$. $\left\{y_1,...,y_n\right\}$ is a set of real values and ${\mathbb{R}}^m:=\left\{({\alpha }_1,...,{\alpha }_m),{\alpha }_j\in \mathbb{R},1\le j\le m\right\}$.