Uniform approximation on compact subsets of the real line

M. B. Korobkova

Matematičeskie zametki, Tome 16 (1974) no. 2, pp. 325-336.

Résumé

Let $\{\varphi\}_{i=0}^n$ be continuous real functions on the compact set $M\subset R$. We consider the problem of best uniform approximation of the function $\varphi_0$ by polynomials $\sum_{i=1}^nc_i\varphi_i$ on $M$. Let $V(\varphi_0,A)$ be a set of polynomials of best approximation on $A\subseteq M$. We show that $V(\varphi_0,M)=\bigcap\limits_{A_{n+1}}V(\varphi_0,A_{n+1})$, where $A_{n+1}$ represents all the possible sets of $n+1$ points $\{x_1,\dots,x_{n+1}\}$ in $M$, containing the characteristic set of the given problem of best approximation and for which the the rank of $\|\varphi_i(x_j)\|$ ($i=1,\dots,n$, $j=1,\dots,n+1$) is equal to $n$. This theorem is applied to a problem of uniform approximation where $\{\varphi_i\}_{i=1}^n$ is a weakly Chebyshev system.