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 -

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  the set of cluster points of the net of best approximations of $f$, $\{P_\epsilon(f)\}$ as $\epsilon \to 0$.