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Pointwise limits of analytic functions. (English) Zbl 1127.30015

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.

MSC:

30E10 Approximation in the complex plane
46J10 Banach algebras of continuous functions, function algebras

Citations:

Zbl 1157.30320
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References:

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