Logarithmic Potential Theory with Applications to Approximation Theory

TL;DR

This paper introduces logarithmic potential theory in the complex plane, highlighting its applications in polynomial and rational approximation, and covers fundamental theorems, concepts, and extensions relevant to approximation theory.

Contribution

It provides a comprehensive introduction to potential theory with applications to approximation, including new insights into external field extensions and fundamental theorems.

Findings

Explains Fekete points, capacity, and Chebyshev constants with examples.

Details key theorems like Frostman's and Riesz Decomposition.

Discusses extensions to external fields in potential theory.

Abstract

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.