This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are based on deep facts from analysis and function theory, such as duality theory and comparison theorems; these are presented in chapters 1 and 3. In keeping with the author's intention to make the book as self-contained as possible, chapter 2 contains an introduction to polynomial and spline approximation. Chapters 4 to 7 apply the theory to specific classes of functions. The last chapter deals with n-widths and generalises some of the ideas of the earlier chapters. Each chapter concludes with commentary, exercises and extensions of results. A substantial bibliography is included. Many of the results collected here have not been gathered together in book form before, so it will be essential reading for approximation theorists.

Table of Contents

Preface; 1. Best approximation and duality in extremal problems; 2. Polynomials and spline-functions as approximating tools; 3. Comparison theorems and inequalities for the norms of functions and their derivatives; 4. Polynomial approximation of classes of functions with bounded r-th derivative in Lp; 5. Spline approximation of classes of functions with bounded r-th derivative; 6. Exact constants in Jackson inequalities; 7. Approximation of classes of functions determined by modulus of continuity; 8. N-widths of functional classes and closely related extremal problems; Appendixes; References.