Inequalities between Best Polynomial Approximants and Smoothness Characteristics of Functions in L2

Abstract

Abstract: Exact constants in Jackson–Stechkin type inequalities are found for the smoothness characteristics (Formula presented.), determined by averaging the norm of finite differences of mth order of functions (Formula presented.). A solution is given of the extremal problem of finding the supremum for best joint polynomial approximations of functions and their successive derivatives on some classes of functions from L2 whose averaged norms of finite differences are bounded above by 1.