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

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: $Γ:=\left\{Γ_1,...,Γ_m\right\}$ is a list of functions where for each $1\leq j\leq m$, $Γ_j:Δ\rightarrow \mathbb R$ with $Δ$ 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 $\mathbb R^m:=\left\{(α_1,...,α_m),\, α_j\in \mathbb R,\, 1\leq j\leq m\right\}$.