TRANSFINITE DIAMETER, ČEBYŠEV CONSTANTS, AND CAPACITY FOR COMPACTA IN $ \mathbf{C}^n$

Abstract

This article considers the higher dimensional analogs of the following classical characteristics of compact planar sets: transfinite diameter, Cebysev constant, and capacity.An affirmative solution is given to the problem, posed by F. Leja in 1957, of whether for the ordinary limit of the sequence defining transfinite diameter exists. The concept of -capacity is introduced, and it is compared with transfinite diameter and another Cebysev constant .For an arbitrary compact set in an analog is considered of a classical theorem of Polya estimating the sequence of Hankel determinants constructed from the coefficients in the power series expansion of an analytic function in a neighborhood of infinity. The estimate comes from the transfinite diameter of the singular set of the function.Bibliography: 10 items.