This book was written for the series Encyclopedia of Mathematics and Its Applications published in the USA and United Kingdom since 1974. A few years ago I was asked to write a monograph Exact Constants in Approximation Theory by the editor of the series Professor G.-C. Rota.

The purpose and objectives of the series were presented by the editorial board as their credo in the following statement:

In their letter to authors the editorial board gives them a free choice of material and style of presentation imposing only two requirements that the monograph should meet, namely: (1) the content should not be of transient significance; and (2) the form of presentation should make the subject matter comprehensible for a wide circle of reades, including non-specialists who may be dealing with other branches of mathematics.

Jackson inequalities (or inequalities of Jackson's type) are expressions in which the approximation error of an individual function is estimated using the modulus of continuity of the function or some of its derivatives. In this context we can consider best approximations either from fixed subspaces or by given methods, and here in particular by linear ones. The only requirement for the approximating function is that the modulus used to estimate the error makes sense for it. The problem of obtaining inequalities which are exact on certain sets can also be posed.

The modulus of continuity is a better characteristic for a function than, for example, its norm in C or Lp and hence obtaining the exact constant in Jackson inequalities needs methods of investigation essentially different from those used in Chapters 4 and 5.

The proofs for exact inequalities of the Jackson type for polynomial and spline approximation are the main content of this chapter. The problem of finding the smallest constants in such inequalities with respect to the whole class of approximating subspaces of fixed dimension will be considered in Section 8.3.

In this chapter we consider polynomial spline approximation. Approximation by splines entered the theory relatively late and immediately acquired a large following; in particular because of their advantages over classical polynomials in the problem of the interpolation of functions. It turns out that, in addition to computational advantage due to the existence of bases with local supports, interpolating splines realize the minimal (for a fixed dimension) deviation for many classes of function. Interpolating polynomials do not even give the best order.

The results in this chapter will in general be connected with two aspects of spline approximation of classes of functions with the rth derivative bounded in Lp: (1) the best approximation by splines with minimal defects; (2) spline interpolation. In the first place we consider those situations when the best approximation is realized by interpolating splines and here it turns out that elementary analysis tools are sufficient if the specific properties of the polynomial splines are used. In Section 5.1 we give some basic broad-based facts to which many problems in spline interpolation can be reduced.

The methods of obtaining exact results in the estimation of the error in best approximation by splines with minimal defects (Section 5.4) are based on essentially different ideas – application of duality relations. These methods allow us to obtain solutions in situations in which the exact estimates for spline interpolation are not known.

The present book gives a presentation of the interval arithmetical tools and methods for the solution of linear and nonlinear systems of equations in the presence of uncertainties in parameters (data errors) and computer arithmetic (rounding errors). It is based on lectures which I gave repeatedly at Freiburg University. A standard background in linear algebra, analysis and numerical analysis is required.

Since there are now over 2000 publications on interval arithmetic (Garloff, 1985, 1987) I have been rather selective in the choice of the material. The major restrictions are:

1. Only finite-dimensional problems are treated; some reasons for this limitation are discussed below. Thus we also do not touch recent applications of interval arithmetic to computer-assisted proofs in analysis (cf. Eckmann & Wittwer, 1985; Lanford, 1986; Matsumoto, Chua & Ayaki, 1988).

2. Eigenvalue problems are not discussed; this omission is mainly due to a lack of time on my part. (For some references see Remarks to Chapter 5.)

3. Range enclosure problems are treated only to the extent needed for an understanding of the basic principles, and to allow applications to implicit functions; the subject essentially belongs to the field of global optimization and a systematic presentation of the state of the art might well fill another book of this size. (For some references see Remarks to Chapter 2.)

In writing this book l tried to achieve several objectives. My first goal was to develop the tools which are necessary to solve the basic problems of finite-dimensional numerical analysis by interval methods.