Smoothness of functions vs. smoothness of approximation processes

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 H"{o}rmander 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 ${\mathbb{T}}^d$, ${\mathbb{R}}^d$, $[-1,1]$, nonlinear wavelet approximation, etc.