Advertisement

Nonlinear Approximation Theory

  • Dietrich Braess

Part of the Springer Series in Computational Mathematics book series (SSCM, volume 7)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Dietrich Braess
    Pages 1-23
  3. Dietrich Braess
    Pages 47-87
  4. Dietrich Braess
    Pages 88-106
  5. Dietrich Braess
    Pages 107-167
  6. Dietrich Braess
    Pages 168-180
  7. Dietrich Braess
    Pages 181-220
  8. Back Matter
    Pages 270-290

About this book

Introduction

The first investigations of nonlinear approximation problems were made by P.L. Chebyshev in the last century, and the entire theory of uniform approxima­ tion is strongly connected with his name. By making use of his ideas, the theories of best uniform approximation by rational functions and by polynomials were developed over the years in an almost unified framework. The difference between linear and rational approximation and its implications first became apparent in the 1960's. At roughly the same time other approaches to nonlinear approximation were also developed. The use of new tools, such as nonlinear functional analysis and topological methods, showed that linearization is not sufficient for a complete treatment of nonlinear families. In particular, the application of global analysis and the consideration of flows on the family of approximating functions intro­ duced ideas which were previously unknown in approximation theory. These were and still are important in many branches of analysis. On the other hand, methods developed for nonlinear approximation prob­ lems can often be successfully applied to problems which belong to or arise from linear approximation. An important example is the solution of moment problems via rational approximation. Best quadrature formulae or the search for best linear spaces often leads to the consideration of spline functions with free nodes. The most famous problem of this kind, namely best interpolation by poly­ nomials, is treated in the appendix of this book.

Keywords

Approximation Interpolation approximation theory

Authors and affiliations

  • Dietrich Braess
    • 1
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochum 1Germany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-61609-9
  • Copyright Information Springer-Verlag Berlin Heidelberg 1986
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-64883-0
  • Online ISBN 978-3-642-61609-9
  • Series Print ISSN 0179-3632
  • Buy this book on publisher's site
Industry Sectors
Aerospace
Oil, Gas & Geosciences
Engineering