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Approximation by random complex polynomials and random rational functions. (English) Zbl 1492.30084

The main purpose of this interesting paper, which deals both with one and several complex variable cases, is to generalize complex approximation theorems (Runge’s theorem, Oka-Weil theorem) to the context of random functions. The authors also prove that the image of a random function over a compact set is a random compact set. They also show that the polynomially and rationally convex hulls of a random compact set are random compact sets. Another result is that the Siciak extremal function and the pluricomplex Green function of a random compact set are random functions.

MSC:

30E10 Approximation in the complex plane
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
30H50 Algebras of analytic functions of one complex variable
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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