Some years ago I regularly gave a traditional course on metric spaces to second-year special honours mathematics students. I was then asked to give a watered-down version of the same material to a class of combined honours students (who were doing several subjects, including mathematics, at a more general level) but, to put it mildly, the course was not a success. It was impossible to motivate students to generalise real analysis when they had never understood it in the first place and certainly could not remember much of it. It was also counter-productive to start the course by revising real analysis because that convinced the students that this was ‘just another analysis course’ and their interest was lost for evermore.

So when I gave the course again the following year I decided to turn the material inside out and to start with the applications (namely the use of contractions in solving a wide range of equations). This meant that the first chapter was a revision of some iterative techniques used to obtain approximations to solutions of equations. This immediately captured the interest of the class: they enjoyed using their calculators and writing programs to solve the equations. Some of the ideas were entirely new to them; for example using iteration to solve an equation with constraints, or solving a differential equation by iterating with an integral and obtaining a sequence of functions.

The second and third chapters were more traditional but the big difference was that the need for distance, function space, closed set, and so on, had been anticipated and motivated. Another difference was that, having approached the subject via iteration, it was then natural to define all the concepts in terms of sequences: hence closed sets (rather than open ones) formed the basis of the approach.

For most students the fourth chapter was the highlight of the course. It consisted of the contraction mapping principle and the use of its algorithmic proof in solving equations.

This is the first in a series of books dealing with approximation and interpolation of functions. Many changes have occurred in this theory during the last decades. In what follows, we shall try to describe some of the problems and achievements of this period.

Until about 1955, the leading force in approximation was the Russians, in particular, Bernstein and his school (Ahiezer), Chebyshev, Kolmogorov, and Markov. The development of the subject in Germany, Hungary, and the United States occurred later. The West certainly leads in the number of papers published—see the bulky Journal of Approximation Theory. The twelve sections that follow review the newer developments.

The two classical books dealing with approximation and interpolation are those of Natanson [0–N] and Ahiezer [0–A]. Important recent books include two Russian works devoted to special problems: Korneichuk [0–K2] (see also [0–K3]) deals with best constants in the trigonometric approximation, while Tihomirov [0–T1] treats extremal problems, particularly widths and optimization. The book of Butzer and Berens [0–B2] introduced functional analytic methods into the field; the two books by de Boor [0–B1] and Schumaker [0-S] deal with splines, an American development rich in practical applications. Karlin and Studden [0–K,] treat Chebyshev systems exhaustively. Books on general approximation theory are those of Rice [0–R], Lorentz [0–L], Dzyadyk [0–D], and Timan [0–T2]; the last book contains a wealth of material. Several books will be mentioned in later sections.