# Analytic capacity

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analytic measure, Ahlfors analytic measure

A set function in the plane, introduced by L. Ahlfors [1], which is an analogue of the logarithmic capacity, and which is suited for the characterization of sets of removable singularities of bounded analytic functions. Let be a bounded closed set in the plane, let be the set of functions which are analytic outside , vanish at infinity and are bounded everywhere outside by the constant 1. Let as . The number , is called the analytic capacity of the set . The analytic capacity of an arbitrary set is usually defined as the supremum of the analytic capacities of its bounded closed subsets.

If is a bounded closed set, then a necessary and sufficient condition in order that each function which is analytic and bounded outside be extendable to , is that the analytic capacity of be zero (the Ahlfors theorem).

There also exist several concepts related to analytic capacity which are suited to the metrics of various other spaces of analytic functions (see, for example, [2] and [3]).

The concept of analytic capacity proved to be well suited to certain problems in approximation theory, where the solution of a number of fundamental problems is formulated in terms of analytic capacity. Thus, [4], any continuous function on a bounded closed set in the plane can be uniformly approximated by rational functions to any desired degree of accuracy, if and only if the equality

holds for any disc of radius .

#### References

 [1] L.V. Ahlfors, "Bounded analytic functions" Duke Math. J. , 14 (1947) pp. 1–11 [2] P.R. Garabedian, "The classes and conformal mapping" Trans. Amer. Math. Soc. , 69 (1950) pp. 392–415 [3] S.O. Sinanyan, "Approximation by analytic functions and polynomials in a real mean" Dokl. Akad. Nauk ArmSSR , 35 : 3 (1962) pp. 107–112 (In Russian) [4] A.G. Vitushkin, "Analytic capacity of sets in problems of approximation theory" Russian Math. Surveys , 22 : 6 (1967) pp. 139–200 Uspekhi Mat. Nauk , 22 : 6 (1967) pp. 141–199