Golubev sums: A theory of extremal problems like the analytic capacity problem and of related approximation processes.

*(English. Russian original)*Zbl 0966.30019
Russ. Math. Surv. 54, No. 4, 753-818 (1999); translation from Usp. Mat. Nauk 54, No. 4, 75-142 (1999).

This is a long paper that develops a theory of extremal problems for complex analytic functions representable by Golubev sums, i.e. functions of the form
\[
f(z)=\sum_{j=1}^{s} \int_{F_j} (\zeta-z)^{-n_j} d\mu_j,\quad z\in G, F=\bigcup_{j=1}^s F_j,
\]
where \(F_j\) are given compact sets in the plane, \(\mu_j\) are measures (complex, real, or positive), \(G\) is the unbounded component of the complement of \(F\), and \(n_j\) are given positive integers. This is a wide class of functions that contains the Cauchy potentials. The extremal problems that the paper studies are generalizations of the classical analytic capacity and Cauchy capacity problems. Their dual problems turn out to be approximation problems. The paper has five sections. \(\S 0\) is a 10-page informative introduction which provides motivation and a summary of the contents. In \(\S 1\) the author proves criteria for the representability of an analytic function by Golubev sums. \(\S 2\) contains a detailed scheme for studying extremal problems. In \(\S 3\) this scheme is implemented for special configurations with complex measures. In \(\S 4\) the author considers Golubev sums with real or positive measures and in \(\S 5\) he studies Golubev sums under Hardy-type conditions (i.e. he requires that certain integral averages be bounded).

Reviewer: Dimitrios Betsakos (Heraklio)

##### MSC:

30C75 | Extremal problems for conformal and quasiconformal mappings, other methods |

30C85 | Capacity and harmonic measure in the complex plane |

31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |

30D10 | Representations of entire functions of one complex variable by series and integrals |

30C70 | Extremal problems for conformal and quasiconformal mappings, variational methods |