Logarithmic Potentials with External Fields

  • Edward B. Saff
  • Vilmos Totik

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 316)

Table of contents

  1. Front Matter
    Pages I-XV
  2. Edward B. Saff, Vilmos Totik
    Pages 1-22
  3. Edward B. Saff, Vilmos Totik
    Pages 23-80
  4. Edward B. Saff, Vilmos Totik
    Pages 81-140
  5. Edward B. Saff, Vilmos Totik
    Pages 141-189
  6. Edward B. Saff, Vilmos Totik
    Pages 191-256
  7. Edward B. Saff, Vilmos Totik
    Pages 257-275
  8. Edward B. Saff, Vilmos Totik
    Pages 277-357
  9. Edward B. Saff, Vilmos Totik
    Pages 359-380
  10. Edward B. Saff, Vilmos Totik
    Pages 381-448
  11. Back Matter
    Pages 449-509

About this book

Introduction

In recent years approximation theory and the theory of orthogonal polynomials have witnessed a dramatic increase in the number of solutions of difficult and previously untouchable problems. This is due to the interaction of approximation theoretical techniques with classical potential theory (more precisely, the theory of logarithmic potentials, which is directly related to polynomials and to problems in the plane or on the real line). Most of the applications are based on an exten­ sion of classical logarithmic potential theory to the case when there is a weight (external field) present. The list of recent developments is quite impressive and includes: creation of the theory of non-classical orthogonal polynomials with re­ spect to exponential weights; the theory of orthogonal polynomials with respect to general measures with compact support; the theory of incomplete polynomials and their widespread generalizations, and the theory of multipoint Pade approximation. The new approach has produced long sought solutions for many problems; most notably, the Freud problems on the asymptotics of orthogonal polynomials with a respect to weights of the form exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the problem of the exact asymptotic constant in the rational approximation of Ixl. One aim of the present book is to provide a self-contained introduction to the aforementioned "weighted" potential theory as well as to its numerous applications. As a side-product we shall also fully develop the classical theory of logarithmic potentials.

Keywords

Potential Potential theory analysis approximation theory capacity external fields extremal points extremum problems logarithmic potentials orthogonal polynomials rational functions weighted polynomials

Authors and affiliations

  • Edward B. Saff
    • 1
  • Vilmos Totik
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of South Florida Institute for Constructive MathematicsTampaUSA
  2. 2.Bolyai InstituteJozsef Attila UniversitySzegedHungary
  3. 3.Department of MathematicsUniversity of South FloridaTampaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-03329-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 1997
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08173-6
  • Online ISBN 978-3-662-03329-6
  • Series Print ISSN 0072-7830
  • Series Online ISSN 2196-9701
  • About this book
Industry Sectors
Electronics
Aerospace
Oil, Gas & Geosciences
Pharma