A New Approach to Derivatives in L2-spaces. (26th July 2020) - Ferreyra, D. E.; Levis, F. E.; Roldán, M. V.
Abstract: In a recent paper of Cuenya and Ferreyra, a condition, namely, the C p -condition in L p -spaces was introduced that is weaker than the notion of L p -derivative given by Calderón-Zygmund. In the present article, we define the Legendre derivative for functions in L 2 generalizing both notions, the C p -condition and L p -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 L 2 by using this new concept of derivative. In addition, we study the convexity of the set of cluster points of the set of best L 2 approximations to a function on a interval when their measures tends to zero.