A New Approach to Derivatives in L2-spaces

Abstract

In a recent paper of Cuenya and Ferreyra, a condition, namely, the Cp-condition in Lp-spaces was introduced that is weaker than the notion of Lp-derivative given by Calderón-Zygmund. In the present article, we define the Legendre derivative for functions in L2 generalizing both notions, the Cp-condition and Lp-derivative in the case p = 2. As a consequence, we give a necessary and sufficient condition for the existence of the best local approximation in L2 by using this new concept of derivative. In addition, we study the convexity of the set of cluster points of the set of best L2 approximations to a function on a interval when their measures tends to zero.