Approximation Theory and Methods

Many excellent books are published on approximation theory and methods. The general texts that are particularly valuable to the present work are the ones by Achieser [2], Cheney [35], Davis [50], Handscomb (ed.) [74], Hayes (ed.) [77], Hildebrand [78], Holland & Sahney [81], Lorentz [100], Rice [132] and [134], Rivlin [138] and Watson [161]. Detailed references and suggestions for further reading are given in this appendix.

Most of the theory in Chapter 1 is taken from Cheney [35] and from Rice [132]. If one prefers an introduction to approximation theory that shows the relations to functional analysis, then the paper by Buck [32] is recommended. We give further attention only in special cases to the interesting problem, mentioned at the end of Section 1.1, of investigating how well any member of B can be approximated from A; a more general study of this problem is in Lorentz [100] and in Vitushkin [160]. The development of the Polya algorithm, which is the subject of Exercise 1.10, into a useful computational procedure is considered by Fletcher, Grant & Hebden [57].

In Chapter 2, as in Chapter 1, much of the basic theory is taken from Cheney [35]. For a further study of convexity the book by Rockafellar [142] is recommended. Several excellent examples of the non-uniqueness of best approximation with respect to the 1- and the ∞-norms are given by Watson [161]. An interesting case of Exercise 2.1, namely when B is the space Rn and the unit ball {f:∥f∥≤1; f∈R} is a polyhedron, is considered by Anderson & Osborne [5].

The point of view in Chapter 3 that approximation algorithms can be regarded as operators is treated well by Cheney [35], and more advanced work on this subject can be found in Cheney & Price [37]. Several references to applications of Theorem 3.1 are given later, including properties of polynomial approximation operators that are defined by interpolation conditions.