Logarithmic Potentials With External Fields

by Saff, E. B.; Totik, Vilmos

• ISBN13: 9783540570783
• ISBN10: 3540570780
• Format: Hardcover
• Copyright: 1997-10-01
• Publisher: Springer Verlag

Summary

This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to signed measures) justify its special attention. The most striking is that it provides a unified approach to seemingly different problems in constructive analysis. These include the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points; the existence and construction of fast decreasing polynomials; the numerical conformal mapping of simply and doubly connected domains; generalization of the Weierstrass approximation theorem to varying weights; and the determination of convergence rates for best approximating rational functions.

Table of Contents

Chapter 0. Preliminaries	1	(22)
0.1 Weak(*) Topology and Lower Semi-continuity	1	(5)
0.2 Fundamentals of Harmonic Functions	6	(3)
0.3 Series Representations of Harmonic Functions	9	(3)
0.4 Poisson's Formula and Applications	12	(6)
0.5 Superharmonic Functions	18	(5)
Chapter I. Weighted Potentials	23	(58)
I.1 The Energy Problem	24	(11)
I.2 Minimum Principle, Dirichlet Problem	35	(8)
I.3 The External Measure	43	(6)
I.4 The Equilibrium Potential	49	(9)
I.5 Fine Topology and Continuity of Equilibrium Potentials	58	(5)
I.6 Weighted Capacity	63	(11)
I.7 Notes and Historical References	74	(7)
Chapter II. Recovery of Measures, Green Functions and Balayage	81	(60)
II.1 Recovering a Measure from Its Potential	83	(14)
II.2 The Unicity Theorem	97	(3)
II.3 Riesz Decomposition Theorem and Principle of Domination	100	(8)
II.4 Green Functions and Balayage Measures	108	(15)
II.5 Green Potentials	123	(14)
II.6 Notes and Historical References	137	(4)
Chapter III. Weighted Polynomials	141	(50)
III.1 Weighted Fekete Points, Transfinite Diameter and Fekete Polynomials	142	(11)
III.2 Where Does the Sup Norm of a Weighted Polynomial Live?	153	(9)
III.3 Weighted Chebyshev Polynomials	162	(7)
III.4 Zero Distribution of Polynomials of Asymptotically Minimal Weighted Norm	169	(8)
III.5 The Function of Leja and Siciak	177	(3)
III.6 Where Does the L(p) Norm of a Weighted Polynomial Live?	180	(7)
III.7 Notes and Historical References	187	(4)
Chapter IV. Determination of the Extermal Measure	191	(66)
IV.1 The Support S(w) of the Extremal Measure	192	(17)
IV.2 The Fourier Method and Smoothness Properties of the Extremal Measure XXX(w)	209	(12)
IV.3 The Integral Equation	221	(6)
IV.4 Behavior of XXX(w)	227	(11)
IV.5 Exponential and Power-Type Functions	238	(7)
IV.6 Circular Symmetric Weights	245	(1)
IV.7 Some Problems from Physics	246	(8)
IV.7.1 Contact Problem of Elasticity	246	(3)
IV.7.2 Distribution of Energy Levels of Quantum Systems	249	(2)
IV.7.3 An Electrostatic Problem for an Infinite Wire	251	(3)
IV.8 Notes and Historical References	254	(3)
Chapter V. Extremal Point Methods	257	(20)
V.1 Leja Points and Numerical Determination of XXX(w)	257	(10)
V.2 The Extremal Point Method for Solving Dirichlet Problems	267	(6)
V.3 The Extermal Point Method for Determining Green Functions and Conformal Mappings	273	(2)
V.4 Notes and Historical References	275	(2)
Chapter VI. Weights on the Real Line	277	(82)
VI.1 The Approximation Problem	278	(23)
VI.2 Approximation with Varying Weights	301	(12)
VI.3 Fast Decreasing Polynomials	313	(13)
VI.4 Discretizing a Logarithmic Potential	326	(8)
VI.5 Norm Inequalities for Weighted Polynomials with Exponential Weights	334	(9)
VI.6 Comparisons of Different Weighted Norms of Polynomials	343	(6)
VI.7 n-Widths for Weighted Entire Functions	349	(3)
VI.8 Notes and Historical References	352	(7)
Chapter VII. Applications Concerning Orthogonal Polynomials	359	(22)
VII.1 Zero Distribution and n-th Root Asymptotics for Orthogonal Polynomials with Exponential Weights	359	(5)
VII.2 Strong Asymptotics	364	(9)
VII.3 Weak(*) Limits of Zeros of Orthogonal Polynomials	373	(6)
VII.4 Notes and Historical References	379	(2)
Chapter VIII. Signed Measures	381	(68)
VIII.1 The Energy Problem for Signed Measures	382	(6)
VIII.2 Basic Theorems for Equilibrium Potentials and Measures Associated with Signed Measures	388	(6)
VIII.3 Rational Fekete Points and a Weighted Variant of a Problem of Zolotarjov	394	(9)
VIII.4 Examples	403	(6)
VIII.5 Rational Approximation of Signum Type Functions	409	(12)
VIII.6 Conformal Mapping of Ring Domains	421	(5)
VIII.7 A Discrepancy Theorem for Simple Zeros of Polynomials	426	(16)
VIII.8 Notes and Historical References	442	(7)
Appendix A. The Dirichlet Problem and Harmonic Measures	449	(16)
A.1 Regularity with Respect to Green Functions	449	(5)
A.2 Regularity with Respect to Dirichlet Problems	454	(4)
A.3 Harmonic Measures and the Generalized Poisson Formula	458	(7)
Appendix B. Weighted Approximation in C(N)	465	(18)
B.1 Pluripotential Theory	466	(5)
B.2 Weighted Polynomials in C(N)	471	(7)
B.3 Fekete Points	478	(2)
B.4 Notes and Historical References	480	(3)
Basic Results of Potential Theory	483	(2)
Bibliography	485	(10)
List of Symbols	495	(6)
Index	501