Transfinite diameter, Chebyshev constants, and capacity for compacta in~$\mathbf C^n$

V. P. Zaharyuta

Sbornik. Mathematics, Tome 25 (1975) no. 3, pp. 350-364

Résumé

This article considers the higher dimensional analogs of the following classical characteristics of compact planar sets: transfinite diameter, Chebyshev constant, and capacity. An affirmative solution is given to the problem, posed by F. Leja in 1957, of whether for $n\geqslant2$ the ordinary limit of the sequence defining transfinite diameter $(d(K)=\varlimsup_{s\to\infty}d_s(K))$ exists. The concept of $\mathbf C^n$-capacity is introduced, and it is compared with transfinite diameter and another Chebyshev constant $T(K)$. For an arbitrary compact set $K\in\mathbf C^n$ 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 titles.