Kolmogorov type inequalities for norms of Riesz derivatives of multivariate functions and some applications

Abstract

Let L∞,s1(ℝm) be the space of functions f ∈ L∞(ℝm) such that ∂f/∂xi ∈ Ls(ℝm) for each i = 1, ...,m. New sharp Kolmogorov type inequalities are obtained for the norms of the Riesz derivatives ∥Dαf∥∞ of functions f ∈ L∞,s1(ℝm). Stechkin’s problem on approximation of unbounded operators Dα by bounded operators on the class of functions f ∈ L∞,s1(ℝm) such that ∥▿f∥s ≤ 1 and the problem of optimal recovery of the operator Dα on elements from this class given with error δ are solved.