Contemporary Mathematics
Vyacheslav Zakharyuta’s Complex Analysis
Christer O. Kiselman
Abstract. The paper gives an account of the work of Vyacheslav Pavlovich
Zakharyuta in the domain of complex analysis, in particular pluripotential
theory, showing the influence of his research during several decades.
1. Introduction
Professor Vyacheslav Pavlovich Zakharyuta has, over many years, made outstand-
ing contributions to mathematics. He has very early found important phenomena
in complex analysis, thereby initiating new roads of research. I shall try here to out-
line some of the most significant of his contributions to analysis in several complex
variables.
I cannot limit myself to an account of his results only; I find it important
to put them into the framework of a more general development of the field of
several complex variables, especially pluripotential theory and the theory of bases
of topological vector spaces of holomorphic functions.
Professor Zakharyuta has also been a very successful advisor of graduate stu-
dents. More than 30 Master Degree students have finished their degree with him
as an advisor, and eleven doctoral theses were successfully defended with him as
an advisor or co-advisor: he has been the principal advisor of S. N. Kadampatta,
N. I. Skiba, P. A. Chalov, N. S. Manzhikova (Nadbitova), Alexander P. Goncharov,
L. V. Runov, V. A. Znamenski˘ ı, M. A. Shubarin, and B. A. Derzhavets. He has
been assistant advisor of Thabet Abdeljavad and Erdal Karapınar.
2. The global extremal function
ozef Siciak introduced (1961, 1962) an extremal function of several complex vari-
ables analogous to the Green function for the unbounded component of the comple-
ment of a compact set in the complex plane and with pole at infinity. He emphasized
that the Green function plays a fundamental role in the theory of interpolation and
approximation of holomorphic functions of one variable by polynomials. Indeed his
2000 Mathematics Subject Classification. Primary 32-03, 32A10, 32A15, 32E30, 32U20,
32U35, 46B15, 01A60, 01A61.
Key words and phrases. Complex analysis in several variables, pluripotential theory, capacity,
global extremal function, relative extremal function, transfinite diameter, bases of topological
vector spaces.
This paper is based on my talk on September 19, 2007, at the Karak¨ oy Communication Center
in
˙
Istanbul, as a part of the conference Functional Analysis and Complex Analysis organized by
Sabancı
¨
Universitesi in honor of Vyacheslav Zakharyuta. It is a pleasure to acknowledge the
help received from Vyacheslav Zakharyuta, Aydın Aytuna, Maciej Klimek, S lawomir Kolodziej,
Alexander Rashkovskii, ozef Siciak, and Ragnar Sigurðsson during the preparation of the present
study.
c XXXX American Mathematical Society
1
Contemporary Mathematics
Volume 481, 2009
c 2009 American Mathematical Society
1
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