A main theme of this book, which lies behind the selection and organization of the material in it, is the use of interpolation in the theory of ordinary differential equations. Inasmuch as different initial andΓor boundary conditions for a differential equation lead naturally to various interpolation techniques to approximate the solution, the book discusses a number of different methods for constructing interpolants. Each of the six chapters features an interpolation method or several related methods and also contains extensive discussion of error estimates and applications. Most applications are to the theory of differential equations, relating to such questions as existence and uniqueness of solutions.
The first four chapters of the book deal with various schemes for constructing algebraic polynomial interpolants. Chapter 1 treats Lidstone interpolation, which is the construction of a polynomial p of degree at most 2n&1 with prescribed values of p (2i) (0) and p (2i) (1), for i=0, ..., n&1. Chapter 2 deals with generalized Hermite interpolation, in which a polynomial p of degree at most n&1 is constructed, using a set of r nodes, with n r; at each node the value of p itself and the values of some number of its consecutive derivatives are prescribed, with the total number of conditions equal to n. These conditions include as special cases classical Hermite interpolation (n=2r, with values of p and p$ given at each point) and classical Lagrange interpolation (n=r). Chapter 3 deals with Abel Goncharov interpolation, that is, the construction of a polynomial p of degree at most n upon points a 0 a 1 } } } a n with preassigned values p (i) (a i ), for i=0, ..., n. Chapter 4 deals with various ``miscellaneous'' interpolation problems which can arise from different types of initial and boundary data.
The last two chapters of the book deal with piecewise polynomial approximation, with the last chapter dealing particularly with spline interpolation. Both of these chapters also include some discussion of approximation of functions of two variables.
In all chapters, the error estimates are carefully constructed and contain actual constants, in many cases best possible. Demonstrably useful in the applications to differential equations, almost all of the error estimates are cast in terms of the norm (usually the uniform norm, sometimes an L p norm) of a high derivative of the function being approximated essentially, the error incurred in an interpolation of n pieces of data is measured by the norm of the nth derivative of the function being approximated. Considering the detail and precision of these estimates, it is perhaps unfortunate that the authors did not address such error estimates which remain useful and valid when the function approximated is not very smooth. But no book can contain everything.
Some particular positive points in the book are found in unexpected places. For example, Section 5.2 (innocuously called ``Preliminaries'') contains some very interesting polynomial and function inequalities for derivatives, including a serious discussion of the constant in the Markov inequality in (unweighted) L 2 [a, b].