Uniform convergence for sequences of best L^{p} approximation

Abstract

Let $f$ be a continuous monotone real function defined on a compact interval $[a,b]$ of the real line. Given a sequence of partitions of $[a,b]$, $% \Delta_n $, $\left\Vert {\Delta }_{n}\right\Vert \rightarrow 0$, and given $l\geq 0,m\geq 1$, let $\mathbf{S}_{m}^{l}(\Delta _{n}) $ be the space of all functions with the same monotonicity of $f$ that are $% \Delta_n$-piecewise polynomial of order $m$ and that belong to the smoothness class $C^{l}[a,b]$. In this paper we show that, for any $m\geq 2l+1$, $\bullet$ sequences of best $L^p$-approximation in $\mathbf{S}_{m}^{l}(\Delta _{n})$ converge uniformly to $f$ on any compact subinterval of $(a,b)$; $\bullet$ sequences of best $L^p$-approximation in $\mathbf{S}_{m}^{0}(\Delta _{n})$ converge uniformly to $f$ on the whole interval $[a,b] $.