Whitney type inequality, pointwise version
Volume 214 / 2013
Studia Mathematica 214 (2013), 167-194
MSC: Primary 26Bxx; Secondary 41A10.
DOI: 10.4064/sm214-2-5
Abstract
The main result of the paper estimates the asymptotic behavior of local polynomial approximation for $L_p$ functions at a point via the behavior of $\mu $-differences, a generalization of the $k$th difference. The result is applied to prove several new and extend classical results on pointwise differentiability of $L_p$ functions including Marcinkiewicz–Zygmund's and M. Weiss' theorems. In particular, we present a solution of the problem posed in the 30s by Marcinkiewicz and Zygmund.
