There are several reasons for studying approximation theory and methods, ranging from a need to represent functions in computer calculations to an interest in the mathematics of the subject. Although approximation algorithms are used throughout the sciences and in many industrial and commercial fields, some of the theory has become highly specialized and abstract. Work in numerical analysis and in mathematical software is one of the main links between these two extremes, for its purpose is to provide computer users with efficient programs for general approximation calculations, in order that useful advances in the subject can be applied. This book presents the view of a numerical analyst, who enjoys the theory, and who is keenly interested in its importance to practical computer calculations. It is based on a course of twenty-four lectures, given to third-year mathematics undergraduates at the University of Cambridge. There is really far too much material for such a course, but it is possible to speak coherently on each chapter for about one hour, and to include proofs of most of the main theorems. The prerequisites are an introduction to linear spaces and operators and an intermediate course on analysis, but complex variable theory is not required.

Spline functions have transformed approximation techniques and theory during the last fifteen years. Not only are they convenient and suitable for computer calculations, but also they provide optimal theoretical solutions to the estimation of functions from limited data. Therefore seven chapters are given to spline approximations. The classical theory of best approximations from linear spaces with respect to the minimax, least squares and L1-norms is also studied, and algorithms are described and analysed for the calculation of these approximations. Interpolation is considered also, and the accuracy of interpolation and other linear operators is related to the accuracy of optimal algorithms. Special attention is given to polynomial functions, and there is one chapter on rational functions, but, due to the constraints of twenty-four lectures, the approximation of functions of several variables is not included.