Introduction

Both Theorems 3.1 and 3.3 (see also Corollary 3.4) provide us with criteria for exactly determining when a given finite-dimensional subspace U of C(K) is a unicity space for C1(K, μ). Nevertheless there are at least two drawbacks to these criteria. In the first place the criteria are generally very difficult, if not impossible, to verify. Except for certain specific subspaces it is almost always impossible to check whether there exist h and u* satisfying the conditions of Theorem 3.1, or whether the inequalities of Corollary 3.4 are always valid. Secondly, the above-mentioned criteria are measure dependent. That is, a specific subspace U may be a unicity subspace for C1(K, μ1), but not for C1(K, μ2), where μ1 and μ2 are both ‘admissible’ measures on K. See Examples 3.1 and 3.3, and Exercise 8 of Chapter 3 where this phenomenon occurs.

It is therefore natural to ask for a characterization of finite-dimensional subspaces U of C(K) with the property that U is a unicity subspace for C1(K, μ) for all ‘nice’ measures μ. Perhaps such a characterization, while it would be interesting in and of itself, might also be easier to verify. Perhaps such a characterization will be an intrinsic property of the subspace U. Perhaps it might be possible to list all such subspaces. This is the problem which we investigate in this chapter.

In Section 2 we delineate a condition called Property A.

Introduction

In the previous three chapters we concerned ourselves with various aspects of the basic problem of best L1-approximation from a finite-dimensional subspace. Some of the topics dealt with were perhaps non-standard, but the setting (i.e., approximation from a finite-dimensional subspace) was a classic one. In this chapter we consider the problem of best one-sided L1-approximation from below, which has been much studied of late. There is an essential difference between these two problems. We are here dealing with approximation from a convex subset of a finite-dimensional subspace which depends on the function being approximated. Moreover the results, i.e., characterization, uniqueness, etc., are different in nature from those of the previous chapters.

To be more precise, let U be a finite-dimensional subspace of C(K). For each f ∈ C(K), we denote by U(f) the convex subset of all u ∈ U satisfying u ≤ f. In Sections 2 and 3 we discuss questions of existence, characterization, and uniqueness in the problem of best approximating f from u(f) in the L1(K, μ)-norm. (We always assume that μ is an ‘admissible’ measure in the sense of Chapter 3.) Somewhat surprisingly, for most ‘reasonable’ U there exist f ∈ C(K) with more than one best approximant (Theorems 5.13 and 5.14).

In Section 4 we ask for exact conditions on the subspace U which imply that there exists a unique best one-sided L1(K, μ) approximant to each f ∈ C(K) from U(f), for a large class of ‘admissible’ μ.

As the title indicates, this appendix contains a collection of facts on T- and WT-systems. It is not our intention to provide a comprehensive treatise on this topic. A more all-inclusive study of T- and WT-systems may be found in Karlin, Studden [1966], Krein, Nudel'man [1977], and Zielke [1979]. Our purpose is to compile, in a cohesive fashion, some of the results used in the previous chapters. Part I contains a few facts concerning T-systems. In Part II we consider WT-systems. In the problem of L1-approximation, WT-systems play a more fundamental role that T-systems (see Chapter IV). As such, the emphasis of this appendix is more on the subject of WT-systems. Appendix B also contains a great deal of material on T- and WT-systems.

Part I. T-Systems

Chebyshev systems (the name was given by Bernstein [1926]) are abbreviated T-systems since at one time the Cyrillic transliteration gave us the spelling Tchebycheff (and variants thereof). They are fundamental in the study of approximation theory. At times T-systems and T-spaces are referred to as Haar systems and Haar spaces, respectively. Formally a system in this context is meant to be a finite sequence of functions, and a space is their linear span. None the less, we shall use the term system for both the system and the space. We will however differentiate between T-systems and Haar systems.

Definition 1. Let B be a compact Hausdorff space and C(B) the set of realvalued continuous functions on B.

Any monograph should speak for itself, and this is no exception. However a few words of explanation would certainly do no harm. The linear theory of best uniform approximation is well documented both in journals and in books. The same cannot be said for the linear theory of best L1-approximation, which has generally received little attention.

This monograph (aside from a few digressions) is about the qualitative theory of best L1-approximation from finite-dimensional subspaces, where the approximation is either two-sided or one-sided. The questions considered are ‘classical’. What is, to me, surprising is that it is only in the last few years that many (but not all) of these questions have been answered. Thus most of the contents of Chapters 4 and 5, as well as some of the contents of Chapters 3 and 6, are the result of very recent research in this area.

This work is not all-encompassing. Various topics which could have been included are not. The most glaring of these omissions is the non-linear theory. I had originally intended to say something about this topic. But I found that I was being forced far a field in order to say relatively little. As of now, nonlinear L1-approximation theory is a poorly developed subject which deserves more attention.

Following each of the first six chapters is a series of exercises. These exercises are an integral part of this work and should be studied. They serve two main functions.

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.