Smoothness of functions versus smoothness of approximation processes

Abstract

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the K-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on 𝕋d, ℝd, [−1, 1], nonlinear wavelet approximation, etc.

Keywords

• measures of smoothness
• k-functionals
• best approximation
• jackson and bernstein inequalities
• littlewood–paley decomposition
• fourier multipliers