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.

In the previous chapters a number of estimates for rational approximation were established. Here we shall be concerned with the exactness of these estimates in the sense of definitions 5.1–5.3 from section 5.1. We use alternance techniques based on some variants of the well-known Chebyshev theorem and Vallée-Poussin theorem for rational approximation.

In section 11.1 there will be given some relatively simple lower bounds, almost all of which are not purely rational in scope. That is, almost all of them are valid for approximation by piecewise monotone functions or piecewise convex functions, particularly for spline approximation. In section 11.2 a non-trivial lower bound is obtained for the rational uniform approximation of functions of bounded variation and given modulus of continuity. Other lower bounds which can be analogously obtained will be omitted.

Some simple lower bounds

In this section we give some relatively elementary lower bounds for rational approximations which are not intrinsically dependent on the nature of the rational functions as an approximating tool. These bounds are based on some more general properties of the rational functions such as piecewise monotony and piecewise convexity.

Negative results for uniform approximation of continuous functions with given modulus of smoothness

In the preceding chapters classes of functions have been found which can be approximated by rational functions better than by polynomials. In this section we show that in the class of all continuous functions with a given modulus of smoothness the rational functions are in general not better than the polynomials as an approximation tool in the uniform metric.

One of the most popular domains in the theory of approximation of functions by means of rational functions is the theory of the Pade approximations. There exist many books and papers which consider this type of approximations. We want only to mention the excellent monograph in two volumes of Baker and Graves-Morris (1981). Here we want to consider some problems connected with the convergence of the Pade approximants, which are not entirely included in that monograph. These results are due to A.A. Gonchar and the group of mathematicians headed by him.

In section 12.1 we give the definition and some promerties of Padé approximants. In section 12.2 we have direct results for the convergence of Pade approximants - the classical theorem of Montessus de Ballore and one of its generalizations, which is due to A.A. Gonchar (1975a). In section 12.3 we give one converse theorem for the convergence of Pade approximants with fixed degree of denominator (the rows of the Pade-table) which is due to Gonchar (unpublished). In section 12.4 we give one more converse theorem of Gonchar connected with the diagonal of the Pade-table. In the notes to the chapter we give some more information about these problems.