Introduction

While this chapter makes no attempt to be exhaustive, nevertheless an appreciable effort has been made to give an idea of the great variety of problems which eventually lead to the solution of one of the two fundamental problems of the Numerical Analysis of Matrices, and in particular those problems whose source lies in approximation of linear partial differential equations arising in Physics.

In this regard, it did not seem fruitless to make some brief observations on the physical aspects of the problems, on their mathematical analysis and, lastly, on their numerical analysis, especially whenever this last depends upon matrix analysis. Thus, in the case of one-dimensional boundary-value problems (introduced in section 3.1), we establish the convergence of the methods of approximation which are considered (cf. Theorems 3.1-2 and 3.4-3).

In sections 3.1, 3.2 and 3.3 some simple examples of boundary-value problems are considered (Poisson's equation, the heat equation and the wave equation). This gives rise to a description offinite-difference methods. Their use, in turn, involves the solution of linear systems with very ‘sparse’ matrices, whose zeros are arranged in a quite remarkable fashion, the matrices involved being examples of tridiagonal or block tridiagonal type. Related matrices which are, moreover, always symmetric are encountered in the variational approximation of these same problems (sections 3.4 and 3.5).

Problems involving the calculation of eigenvalues and eigenvectors of matrices are encountered in the approximation of the ‘stationary solutions’, or ‘waves’ of various physical systems. It is shown, in this respect, in section 3.6, that these problems very often make their appearance in the form of ‘generalised’ eigenvalue problems.

Other sources of linear systems are the problems of the interpolation of data by means of'piecewise polynomial’ functions, notably spline functions, as well as problems of least-squares approximation: these problems are encountered in section 3.7.

Finally, we mention some other examples, which get further treatment later in this work:

the solution of non-linear systems of equations, where the use of Newton's method leads to the solution of a sequence of'tangential’ systems of linear equations (section 7.5);

problems of the minimisation of a quadratic functional with linear constraints (section 7.2);

the implementation of Uzawa's method (section 9.4);

the implementation of the simplex method (section 10.3).

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.