We shall begin with a short survey of the basic results related to linear approximations (i.e. approximation by means of linear subspaces) so that one can feel better the peculiarities, the advantages as well as some shortcomings of the rational approximation. In this chapter we shall consider the problems of existence, uniqueness and characterization of the best approximation (best polynomial approximation). At the end of the chapter we shall consider also numerical algorithms for finding the best uniform polynomial approximation.

Approximation in normed linear spaces

Let X be a normed linear space. Recall that X is said to be a normed linear space if:

1. (i) X is a linear space, i.e. for its elements sum, and product with real numbers, are defined so that the standard axioms of commutativity and associativity are satisfied;

2. (ii) X is a normed space, i.e. to each x ϵ X there corresponds a nonnegative real number ∥x∥ satisfying the axioms

1. (a) ∥x∥ ≥ 0, ∥x∥ = 0 iff x = 0,

2. (b) ∥λx∥ = |λ| ∥x∥, λ a real number,

3. (c) ∥x + y∥ ≤ ∥x∥ + ∥y∥ (the triangle inequality).

The most essential problems in the qualitative theory of the best approximation are the problems of existence, uniqueness and characterization of the best approximation. Finally the problems connected with the continuity of the operator of the best approximation, or, as is mainly used, the continuity of the metric projection, are considered. In this chapter we shall consider these questions for the best rational approximation. The difficulties arise from the fact that the set Rnm of all rational functions of order (n, m) (see the exact definition in section 2.1) is not a finite dimensional linear space and the bounded sets in Rnm are not compact in C[a, b] or in Lp(a, b). Nevertheless we shall see that there exists an element of best approximation in C[a, b] and Lp(a,b) (section 2.1). Moreover in C[a, b] we have uniqueness and characterization of the best approximation by means of an alternation, as in the linear case (see section 2.2). Unfortunately in Lp(a,b), 1 ≤ p < ∞, we do not have uniqueness (section 2.3). In section 2.4 we consider the problem of continuity of the metric projection in C[a, b] – the metric projection is continuous only in the so-called ‘normal points’ (see section 2.4). In section 2.5 we consider numerical methods for obtaining the rational function of best uniform approximation. We should like to remark that we examine only the usual rational approximation.

Rational functions are a classical tool for approximation. They turn out to be a more convenient tool for approximation in many cases than polynomials which explains the constant increase of interest in them. On the other hand rational functions are a nonlinear approximation tool and they possess some intrinsic peculiarities creating a lot of difficulties in their investigation. After the classcial results of Zolotarjov from the end of the last century substantial progress was achieved in 1964 when D. Newman showed that |x| is uniformly approximated by rational functions much better than by algebraic polynomials.

Newman's result stimulated the appearance of many substantial results in the field of rational real approximations. Our aim in this book is to present the basic achievements in rational real approximations. Nevertheless, for the sake of completeness we have included some results referring to the field of complex rational approximations in Chapters 6 and 12. Also, in order to stress some peculiarities of rational approximations we have included for comparison some classical and more recent results from the linear theory of approximation. On the other hand, since rational approximations are closely connected with spline approximations, we have included as well some results concerning spline approximations.

In this chapter we shall consider rational approximation of functions with respect to the Hausdorff distance. The Hausdorff distance in the space C[a, b] of the continuous functions in the interval [a, b] was introduced by Bl. Sendov and B. Penkov (1962). After this Bl. Sendov developed the theory of approximation of bounded functions by means of algebraic polynomials with respect to the Hausdorff distance. Many mathematicians have obtained results in the theory of approximation of functions with respect to the Hausdorff distance – the results are collected in the book of Bl. Sendov (1979).

In section 9.1 we give the definition of Hausdorff distance in the set of all bounded functions in a given interval and we consider some of its properties.

In section 9.2 we consider the most interesting examples of rational approximation in Hausdorff distance – rational approximation of sign x. In our opinion this result is basic in the theory of rational approximation – from here follows the most essential results for uniform and Lp rational approximation – for example Newman's result for |x|. The Hausdorff distance is the natural distance by means of which we can explain the fact that sign x can be approximated to order O(e−c√n) by means of rational functions.

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.