A New Approach to Derivatives in L2-spaces. (26th July 2020)

• Record Type: Journal Article
• Authors: Ferreyra, D. E.; Levis, F. E.; Roldán, M. V.
• Is Part Of: Numerical functional analysis and optimization. Volume 41:Number 10(2020)
• Page Start: 1272
• Page End: 1285
• Publication Date: 2020-07-26
• Subjects: Best local approximation -- L2-spaces -- Legendre derivative; 26A24 -- 41A10 -- 43A15; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2020.1738459
• Languages: English
• ISSNs: 0163-0563

Abstract

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.