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Logarithmic potentials with external fields. (English) Zbl 0881.31001
Grundlehren der Mathematischen Wissenschaften. 316. Berlin: Springer. xv, 505 p. (1997).
Let \(M(E)\) denote the set of all probability measures on a compact set \(E\) in the complex plane. The equilibrium distribution \(\mu_E\) is the unique element of \(M(E)\) for which the energy \[ I(\mu)= \iint\log (1/|z-t|)d\mu(z) d\mu(t) \] is minimal among all \(\mu\in M(E)\). The determination of \(\mu_E\) is a classical problem in potential theory, and the measure \(\mu_E\) arises naturally in other areas of analysis; for example, it describes the limiting behaviour of Fekete points, which are important in polynomial interpolation.
The book under review is concerned with a generalization of the equilibrium distribution and especially with numerous recent applications to various problems in constructive analysis. Let \(w\) be a weight function on \(E\) (that is, \(w\) is non-negative and upper semicontinuous on \(E\) and the set where \(w>0\) has positive logarithmic capacity). The weighted energy of \(\mu\in M(E)\) is \[ I_w(\mu)=\iint \log(1/|z-t|w(z)w(t))d\mu(z) d\mu(t)= I(\mu)+2\int Q d\mu, \] where \(Q=-\log w\). There is a unique \(\mu_w\in M(E)\) for which \(I_w(\mu)\) attains a minimum value. In terms of electrostatics, \(\mu_w\) is the equilibrium distribution of a unit charge on \(E\) in the presence of an external field \(Q\). The external field problem (that of determining \(I_w(\mu_w)\) and \(\mu_w\)) originates in work of Gauss and was investigated by O. Frostman and by F. Leja and his school during the period 1935-60. Interest in the problem was renewed in the 1980s when potentials with external fields were used to study polynomials orthogonal with respect to exponential weights on the real line. For this and other applications, it is important to note that the boundedness of \(E\) can be dispensed with, provided that in the unbounded case we assume \(|z|w(z)\to 0\) as \(|z|\to+\infty\), \(z\in E\).
A preliminary chapter reviews prerequisite material. Then comes a detailed study of the external field problem. The development of the theory requires several fundamental results from classical plane potential theory, such as generalized maximum principles, the Riesz decomposition theorem, the domination principle, and Evans’ theorem; these are presented and proved as they are needed, and Wiener’s regularity criterion and the Dirichlet problem are treated in an appendix. A major topic of investigation is the support \(S_w\) of the extremal measure \(\mu_w\). In contrast with the classical unweighted \((w\equiv 1)\) case, in which \(\mu_E\) is always supported on the outer boundary of \(E\), the support \(S_w\) can be quite a general subset of \(E\) (depending on \(w\)), though \(S_w\) must be compact, even when \(E\) is not.
A large part of the book is devoted to a wide variety of applications to problems in constructive analysis. The topics treated include: asymptotic analysis of polynomials orthogonal with respect to a weight function on an unbounded interval (for example, \(w(x)= \exp(-|x|^\alpha)\), \(\alpha>0\) on \((- \infty,+\infty)\)); asymptotic behaviour as \(n\to\infty\) of weighted Fekete points that maximize \(\prod_{1\leq j<k\leq n}|z_j-z_k|w(z_j)w(z_k)\) among all possible choices of \(z_1,\dots, z_n\) in a closed set \(E\); fast decreasing polynomials – for a prescribed \(\varphi :[-1,1]\to [0,+\infty)\) one seeks polynomials \(p_n\) of degree \(n\) such that \(p_n(0)=1\) and \(|p_n|\leq e^{-n\varphi}\) on \([-1,1]\); incomplete polynomials of the form \(\sum_{k=s}^n a_kx^k\), where \(s\geq \theta n\) for some \(\theta>0\); conformal mappings of simply and doubly connected domains; Weierstrass-type theorems involving uniform approximation by functions \(w^np_n\), where \(p_n\) is a polynomial of degree \(n\); rates of convergence of best rational approximants to certain functions \(f\) (for example, \(f(x)= e^{-x}\) on \([0,+\infty)\)); modelling of certain elasticity problems.
The external field problem is also related to important concepts within potential theory. For instance, if \(U^\mu\) denotes the logarithmic potential of a measure \(\mu\), then on each bounded component of the complement of \(S_w\), the potential \(U^{\mu_w}\) is, apart from an additive constant, the solution of the Dirichlet problem for boundary data \(\log w\). To take another example, the balayage of a measure \(\nu\) to a compact set \(E\) is the extremal measure corresponding to the external field \(Q=-U^\nu\) on \(E\).
The book includes a final chapter on the case where the measure \(\mu\) is allowed to be signed and an appendix by T. Bloom about weighted approximation in \(\mathbb{C}^N\), which uses pluripotential theory.
Each chapter ends with a section of notes and historical references which includes discussion and citations for many of the theorems. However, a large number of results and proofs appear for the first time in the book.
This is a very well written and authoritative work by two of the subject’s leading researchers.

MSC:
31-02 Research exposition (monographs, survey articles) pertaining to potential theory
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30C85 Capacity and harmonic measure in the complex plane
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
42A50 Conjugate functions, conjugate series, singular integrals
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