A New Approach to Derivatives in L2-Spaces

D. E. FERREYRA; F.E. LEVIS; M.V. ROLDÁN

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION

TAYLOR & FRANCIS INC

Lugar: Londres; Año: 2020 vol. 41 p. 1272 - 1285

0163-0563

In a recent paper cite{CF} 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 Calder´on-Zygmund cite{CZ}. 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.