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 $[a,b]$, , $\Vert {\Delta }_n\Vert \to 0$, and given $l\ge 0,m\ge 1$, let ${\mathbf{S}}_m^l({\Delta }_n)$ be the space of all functions with the same monotonicity of $f$ that are -piecewise polynomial of order $m$ and that belong to the smoothness class $C^l[a,b]$. In this paper we show that, for any $m\ge 2l+1$, $\bullet$ 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 $[a,b]$.

16 pages, not submitted to any journal at the moment