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Logarithmic Potentials with External Fields : Grundlehren der mathematischen Wissenschaften - Edward B. Saff

Logarithmic Potentials with External Fields

Grundlehren der mathematischen Wissenschaften

Hardcover Published: 9th October 1997
ISBN: 9783540570783
Number Of Pages: 505

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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.

Preliminariesp. 1
Weighted Potentialsp. 23
Recovery of Measures, Green Functions and Balayagep. 81
Weighted Polynomialsp. 141
Determination of the Extremal Measurep. 191
Extremal Point Methodsp. 257
Weights on the Real Linep. 277
Applications Concerning Orthogonal Polynomialsp. 359
Signed Measuresp. 381
The Dirichlet Problem and Harmonic Measuresp. 449
Weighted Approximation in C[superscript N]p. 465
Basic Results of Potential Theoryp. 483
Bibliographyp. 485
List of Symbolsp. 495
Indexp. 501
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540570783
ISBN-10: 3540570780
Series: Grundlehren der mathematischen Wissenschaften
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 505
Published: 9th October 1997
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 2.02