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

We study analogues of analytic capacity for classes of analytic functions representable via some special analytic machinery, which we refer to as “Golubev sums”. A Golubev sum contains derivatives of various (given) orders of Cauchy potentials (in particular, the Cauchy potentials themselves can occur in a Golubev sum). Furthermore, the measures determining distinct terms of a Golubev sum are in general defined on distinct compact sets. We consider Golubev sums with various types of measures: complex, real, and positive. We present an abstract scheme for studying extremal problems like the analytic capacity problem. The dual problems turn out to be approximation problems in which the size of the approximants is taken into account. In the case of positive measures, the approximation problem is transformed into a problem in which one has to move a given element of a space into a given cone in that space by adding linear combinations of elements of a given subspace with coefficients as small as possible. As a preliminary, we state criteria for the representability of an analytic function by Golubev sums of various kinds. These criteria generalize known criteria for representability by Cauchy potentials.