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Logarithmic Potential Theory with Applications to Approximation Theory

Author(s): E. B. Saff

Journal: Surveys in Approximation Theory
ISSN 1555-578X

Volume: 5;
Start page: 165;
Date: 2010;
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Keywords: Logarithmic potential | Polynomial approximation | Rational approximation | Transfinite diameter | Capacity | Chebyshev constant | Fekete points | Equilibrium potential | Superharmonic functions | Subharmonic functions | Green functions | Rates of polynomial and rational approximation | Condenser capacity | External fields

We provide an introduction to logarithmic potential theory in the complex plane that particularly emphasizes its usefulness in the theory of polynomial and rational approximation. The reader is invited to explore the notions of Fekete points, logarithmic capacity, and Chebyshev constant through a variety of examples and exercises. Many of the fundamental theorems of potential theory, such as Frostman’s theorem, the Riesz Decomposition Theorem, the Principle of Domination, etc., are given along with essential ideas for their proofs. Equilibrium measures and potentials and their connections with Green functions and conformal mappings are presented. Moreover, we discuss extensions of the classical potential theoretic results to the case when an external field is present.

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