A Lebesgue type differentiation theorem for best approximations by constants in Orlicz spaces

Abstract

The best approximation operator by constants is extended from an Orlicz space $L^{\varphi}({\mathbb R}^m)$ to the space $L^{\varphi '} ({\mathbb R}^m),$ and some properties of this extended operator are established. Let $f_{\varepsilon}(x)$ be any best approximation of $f\in L^{\varphi '} ({\mathbb R}^m)$ on a suitable set $\beta_{\varepsilon} (X) \subset \mathbb{ R}^m.$ Weak and strong inequalities are proved for the maximal function associated with the family $\{f_{\varepsilon}(x)\}$ which are used in the study of pointwise convergence of $f_{\varepsilon}(x)$ to $f(x).$