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Harmonic approximation. (English) Zbl 0826.31002

London Mathematical Society Lecture Note Series. 221. Cambridge: Univ. Press. xiii, 132 p. (1995).
The main purpose of the book under review is to give a coherent account of results obtained during a recent period of rapid development in the theory of harmonic approximation. The book also includes details of classical results and a number of attractive applications of the theory. The author has been the main contributor to this area of research during the last few years.
Given a non-empty subset of \(\mathbb{R}^n\) \((n\geq 2)\), let \({\mathcal H} (F)\) denote the space of functions that are harmonic on (an open set containing) \(F\). Let \(E\) be a relatively closed proper subset of an open set \(\Omega\) in \(\mathbb{R}^n\). A central problem is to determine conditions on the pair \((\Omega, E)\) for every function in \({\mathcal H} (E)\) (or \(C(E)\cap {\mathcal H} (E^0))\) to be approximable on \(E\) (uniformly or in some stronger sense) by functions in \({\mathcal H} (\Omega)\).
Chapter 1 treats this problem in the case where \(E\) is compact. The results extend classical work which dealt with the special case where \(\Omega= \mathbb{R}^n\).
The first work to deal with fairly general non-compact sets \(E\) appeared in the early 1980s and treated uniform approximation. Subsequent research concerned the following stronger types of approximation; \({\mathcal F} (E)\) denotes either \({\mathcal H} (E)\) or \(C(E) \cap {\mathcal H} (E^0)\).
(i) Tangential approximation: for every \(u\in {\mathcal F} (E)\) and every function \(s\) that is positive and superharmonic on an open set containing \(E\) there exists \(v\in {\mathcal H} (\Omega)\) such that \(|u-v |<s\) on \(E\).
(ii) Carleman approximation: for every \(u\in {\mathcal H} (E)\) and every continuous function \(\varepsilon: E\to (0, 1]\) there exists \(v\in {\mathcal H} (\Omega)\) such that \(0< u-v< \varepsilon\) on \(E\).
(iii) Tangential approximation at infinity: for every \(u\in {\mathcal H} (E)\) and all positive numbers \(\varepsilon\) and \(a\) there exists \(v\in {\mathcal H} (\Omega)\) such that \(|u(x)- v(x)|< \varepsilon (1+|x|)^{-a}\) for all \(x\in E\).
Recent work has culminated in the characterization of pairs \((\Omega, E)\) for which each of these three types of approximation is possible for both \({\mathcal F} (E)= {\mathcal H} (E)\) and \({\mathcal F} (E)= C(E)\cap {\mathcal H} (E^0)\). Chapters 3, 4, 5 are devoted to these types of approximation respectively. Preliminary results on harmonic fusion, a technique for passing from approximation on compact sets to approximation on non- compact ones, is the subject of Chapter 2.
Chapter 6 concerns recent work on superharmonic extension and approximation. In Chapter 7 results on tangential approximation are used to characterize unbounded open sets for which (a natural form of) the Dirichlet problem is always solvable. This work provides a solution to a long-standing problem.
The final chapter gives a miscellany of applications of harmonic approximation. Notably, constructions are given for a non-trivial harmonic function on \(\mathbb{R}^n\) with zero integral on every \((n-1)\)- dimensional hyperplane and for a universal harmonic function (that is, one whose translates are dense in the space of all harmonic functions on \(\mathbb{R}^n\), topologized by local uniform convergence).
The book is well written: the organization is skilful; some proofs are new; analogies with the theory of holomorphic approximation are clearly explained. Notes on each chapter indicate relevant literature. The essential facts from classical potential theory that are required are clearly set out in a preliminary Chapter 0.
The book is timely and authoritative, and is strongly recommended to anyone with an interest in harmonic approximation or related fields.

MSC:

31-02 Research exposition (monographs, survey articles) pertaining to potential theory
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
41A30 Approximation by other special function classes
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