TY - JOUR
T1 - $C^p$ Condition and the Best Local Approximation
JO - Analysis in Theory and Applications
VL - 1
SP - 58
EP - 67
PY - 2017
DA - 2017/01
SN - 31
DO - http://doi.org/10.4208/ata.2015.v31.n1.5
UR - https://global-sci.org/intro/article_detail/ata/4622.html
KW - Best $L^p$ approximation, local approximation, $L^p$ differentiability.
AB - <p style="text-align: justify;">In this paper, we introduce a condition weaker than the $L^p$ differentiability, which we call $C^p$ condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be $L^p$ differentiable. In addition, we study the convexity of &nbsp;the set of cluster points of the net of best approximations of $f$, $\{P_\epsilon(f)\}$ as $\epsilon \to 0$.</p>