Davie, A. M. Pointwise limits of analytic functions. (English) Zbl 1127.30015 J. Lond. Math. Soc., II. Ser. 75, No. 1, 133-145 (2007). This interesting paper was prompted by the following question. Under what conditions can a given complex-valued function, \(f\), defined on an open subset \(D\) of \(\mathbb C\), be expressed as the pointwise limit of a sequence of analytic functions on \(D\)? After a nice summary of some of the known results (cf. Introduction), the author establishes a necessary and sufficient condition in Section 1. In Section 2, the author shows how part of the theory can be developed in a very general setting in which analytic functions are replaced by an algebra, \(A\), of functions defined on a set \(X\). In this general context, the author is also able to characterise analogues of the higher Baire classes. Many of the arguments presented depend on Runge’s theorem. Reviewer: George Csordas (Honolulu) Cited in 1 ReviewCited in 1 Document MSC: 30E10 Approximation in the complex plane 46J10 Banach algebras of continuous functions, function algebras Keywords:Pointwise approximation of analytic functions; algebra of functions; Baire classes Citations:Zbl 1157.30320 PDFBibTeX XMLCite \textit{A. M. Davie}, J. Lond. Math. Soc., II. Ser. 75, No. 1, 133--145 (2007; Zbl 1127.30015) Full Text: DOI References: [1] Beardon, On the pointwise limit of complex analytic functions, Amer. Math. Monthly 110 pp 289– (2003) · Zbl 1187.30003 · doi:10.2307/3647878 [2] Davidson, Pointwise limits of analytic functions, Amer. Math. Monthly 90 pp 391– (1983) · Zbl 0518.30003 · doi:10.2307/2975578 [3] Gamelin, Uniform algebras (1969) [4] Gardiner, Pointwise convergence and radial limits of harmonic functions, Israel J. Math 145 pp 243– (2005) · Zbl 1082.31001 · doi:10.1007/BF02786692 [5] Kuratowski, Topology 1 (1966) [6] Lavrentieff, Sur les fonctions d’une variable complexe représentables par des séries de polynomes (1936) · JFM 62.1205.01 [7] Mergelyan, Uniform approximation to functions of a complex variable, Uspehi Mat. Nauk 7 pp 31– (1952) · Zbl 0059.05902 [8] Newman, Elements of the topology of plane sets of points (1954) [9] Osgood, Note on the functions defined by infinite series whose terms are analytic functions of a complex variable; with corresponding theorems for definite integrals, Ann. of Math. III ((2)) pp 25– (1901) · JFM 32.0399.01 · doi:10.2307/1967630 [10] Runge, Zur Theorie der eindeutigen analytischen Funktionen, Acta Math 6 pp 228– (1885) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.