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SAT 5 (2010), 2
Surveys in Approximation Theory, 5 (2010), 165-200.
Logarithmic Potential Theory with
Applications to Approximation Theory
E. B. Saff
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.
E-print: arXiv:1010.3760
Published: 12 October 2010.
Edward B. Saff
Center for Constructive Approximation
Department of Mathematics
Vanderbilt University
Nashville TN 37240 USA
E-mail: edward.b.saff@vanderbilt.edu
Homepage: http://www.math.vanderbilt.edu/~esaff