$C^p$ Condition and the Best Local Approximation

Abstract

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 $\epsilon \to 0$.