A minimax formula for the best natural C([0,1])-approximate by nondecreasing functions.

Abstract

Let $f$ be a function in $C([0,1])$. We denote by $f_p$ the best approximant to $f$ in $L_p([0,1])$ by nondecreasing functions. It is well known that the limit $f_{*}:=\lim_{p\to\infty}f_p$ exists and $f_{*}$ is a best approximant to $f$ in $C([0,1])$ by nondecreasing functions. In this paper we show an explicit formula for the function $f_{*}$ and we prove some additional minimization properties of $f_{*}$.