Geodesic
Transfinite diameter, Chebyshev constants, and capacity for compacta in $\mathbf C^n$
Sbornik. Mathematics, Tome 25 (1975) no. 3, pp. 350-364 
V. P. Zaharyuta. Transfinite diameter, Chebyshev constants, and capacity for compacta in $\mathbf C^n$. Sbornik. Mathematics, Tome 25 (1975) no. 3, pp. 350-364. http://geodesic.mathdoc.fr/item/SM_1975_25_3_a1/
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     title = {Transfinite diameter, {Chebyshev} constants, and capacity for compacta in~$\mathbf C^n$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, izd-vo «Nauka», Moskva, 1966 | MR

[2] F. Leja, “Problems à resondre poses a la conference”, Colloq. Math., 7 (1959), 153 | MR

[3] F. Leja, Theory of analytic functions, Warsaw, 1957 | MR

[4] M. Schieffer, J. Siciak, “Transfinite diameter and analitic continuation of function of two complex variables”, Analysis and related topics, Studies Math., Stanford, 1962, 341–358 | MR

[5] V. P. Sheinov, Analogi teoremy Polia i ratsionalnye funktsii mnogikh peremennykh, dep. 7. 575, In-t fiziki Sib. otd. AN SSSR, Krasnoyarsk, 1968

[6] L. I. Ronkin, Vvedenie v teoriyu tselykh funktsii mnogikh peremennykh, izd-vo «Nauka», Moskva, 1971 | MR

[7] N. I. Akhiezer, Lektsii po teorii approksimatsii, izd-vo «Nauka», Moskva, 1965 | MR

[8] V. P. Zakharyuta, “Teoremy Bernshteina o nailuchshikh polinomialnykh priblizheniyakh dlya golomorfnykh i garmonicheskikh funktsii mnogikh peremennykh”, Tezisy dokladov na Vsesoyuznoi konferentsii po TFKP, Kharkov, 1971, 80–81

[9] B. A. Fuks, Vvedenie v teoriyu analiticheskikh funktsii mnogikh kompleksnykh peremennykh, Fizmatgiz, Moskva, 1962

[10] H. G. Tillmann, “Randverteilungen analytischer Funktionen und Distributionen”, Math. Z., 59:1 (1953), 61–83 | DOI | MR