Exact Constants in Approximation Theory [Paperback]

A self-contained introduction for non-specialists, or a reference work for experts, on the area of approximation theory concerned with exact constants.This book provides a self-contained introduction 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. A wide range of questions and problems are discussed.This book provides a self-contained introduction 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. A wide range of questions and problems are discussed.This book is a self-contained introduction to the particular area of approximation theory 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; and optimal reconstruction of functions and linear functionals. Many of the results are based on facts from analysis and function theory, such as duality theory and comparison theorems. Each chapter concludes with commentary, exercises, and extensions of results, and a substantial bibliography is also included.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. Approxl|