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Logarithmic Potentials with External Fields

  • Book
  • © 1997

Overview

Authors:
  1. Edward B. Saff

    1. Department of Mathematics, University of South Florida Institute for Constructive Mathematics, Tampa, USA

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  2. Vilmos Totik

    1. Bolyai Institute, Jozsef Attila University, Szeged, Hungary
      Department of Mathematics, University of South Florida, Tampa, USA

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • This book is devoted to the systematic mathematical study of the equilibrium problem of potential theory under the influence of an external field; as well as to its many-fold applications.
  • The presentation not only serves as an introduction to the classical theory of logarithmic potentials, it provides a bridge to the frontiers of current research.

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

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-XV
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  2. Preliminaries

    • Edward B. Saff, Vilmos Totik
    Pages 1-22

  3. Weighted Potentials

    • Edward B. Saff, Vilmos Totik
    Pages 23-80

  4. Recovery of Measures, Green Functions and Balayage

    • Edward B. Saff, Vilmos Totik
    Pages 81-140

  5. Weighted Polynomials

    • Edward B. Saff, Vilmos Totik
    Pages 141-189

  6. Determination of the Extremal Measure

    • Edward B. Saff, Vilmos Totik
    Pages 191-256

  7. Extremal Point Methods

    • Edward B. Saff, Vilmos Totik
    Pages 257-275

  8. Weights on the Real Line

    • Edward B. Saff, Vilmos Totik
    Pages 277-357

  9. Applications Concerning Orthogonal Polynomials

    • Edward B. Saff, Vilmos Totik
    Pages 359-380

  10. Signed Measures

    • Edward B. Saff, Vilmos Totik
    Pages 381-448

  11. Back Matter

    Pages 449-509
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, University of South Florida Institute for Constructive Mathematics, Tampa, USA

    Edward B. Saff

  • Bolyai Institute, Jozsef Attila University, Szeged, Hungary

    Vilmos Totik

  • Department of Mathematics, University of South Florida, Tampa, USA

    Vilmos Totik

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