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

TL;DR

This paper investigates the problem of optimally approximating finite data sets using linear combinations of functions by formulating it as a linear programming problem, enabling efficient computation of the best uniform approximation.

Contribution

It introduces a linear programming approach to solve the uniform approximation problem for finite sets with arbitrary functions, extending previous methods.

Findings

Linear programming efficiently computes best uniform approximation.

The method applies to arbitrary functions and data sets.

Provides theoretical foundation for approximation using linear combinations.

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 $\mathbb R^m:=\left\{(\alpha_1,...,\alpha_m),\, \alpha_j\in \mathbb R,\, 1\leq j\leq m\right\}$.