The most essential question in the quantitative theory of approximation is the connection between the degree of the best approximation to a given function f by means of some tool for approximation (algebraic polynomials, trigonometric polynomials, rational functions, spline functions and others) with respect to a given metric (uniform, Lpand others) and the smoothness properties of f (differentiability, Lipschitz conditions etc.).

The solutions of these questions in linear approximations usually use the moduli of continuity and smoothness. So we shall begin in section 3.1 with some definitions and properties of the moduli of smoothness in C[a, b] and in Lp[a, b]. In section 3.2 and 3.3 we give the classical theorems of Jackson and Bernstein for best trigonometrical Lp approximation. In section 3.4 we consider briefly the best approximation by means of algebraical polynomials in [–1,1] and the singularities connected with them. Finally in section 3.5 we consider the K-functional of J. Peetre, which is the abstract version of the moduli of smoothness, and its application for the characterization of the degree of the best approximation in the abstract case, using abstract Jackson type and Bernstein type theorems.