Exact Constants in Approximation Theory (Encyclopedia of Mathematics and its Applications, Series Number 38)

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. 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. N. Korneichuk ; Translated From The Russian By K. Ivanov. Translation Of: Tochnye Konstanty V Teorii Priblizhenii︠a︡. Includes Bibliographical References. 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.