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Integral representations of holomorphic functions play an important
part in the classical theory of functions of one complex variable
and in multidimensional com plex analysis (in the later case,
alongside with integration over the whole boundary aD of a domain D
we frequently encounter integration over the Shilov boundary 5 =
S(D)). They solve the classical problem of recovering at the points
of a do main D a holomorphic function that is sufficiently
well-behaved when approaching the boundary aD, from its values on
aD or on S. Alongside with this classical problem, it is possible
and natural to consider the following one: to recover the
holomorphic function in D from its values on some set MeaD not
containing S. Of course, M is to be a set of uniqueness for the
class of holomorphic functions under consideration (for example,
for the functions continuous in D or belonging to the Hardy class
HP(D), p ~ 1).
Plurisubharmonic functions playa major role in the theory of
functions of several complex variables. The extensiveness of
plurisubharmonic functions, the simplicity of their definition
together with the richness of their properties and. most
importantly, their close connection with holomorphic functions have
assured plurisubharmonic functions a lasting place in
multidimensional complex analysis. (Pluri)subharmonic functions
first made their appearance in the works of Hartogs at the
beginning of the century. They figure in an essential way, for
example, in the proof of the famous theorem of Hartogs (1906) on
joint holomorphicity. Defined at first on the complex plane IC, the
class of subharmonic functions became thereafter one of the most
fundamental tools in the investigation of analytic functions of one
or several variables. The theory of subharmonic functions was
developed and generalized in various directions: subharmonic
functions in Euclidean space IRn, plurisubharmonic functions in
complex space en and others. Subharmonic functions and the
foundations ofthe associated classical poten tial theory are
sufficiently well exposed in the literature, and so we introduce
here only a few fundamental results which we require. More detailed
expositions can be found in the monographs of Privalov (1937),
Brelot (1961), and Landkof (1966). See also Brelot (1972), where a
history of the development of the theory of subharmonic functions
is given."
Integral representations of holomorphic functions play an important
part in the classical theory of functions of one complex variable
and in multidimensional com plex analysis (in the later case,
alongside with integration over the whole boundary aD of a domain D
we frequently encounter integration over the Shilov boundary 5 =
S(D)). They solve the classical problem of recovering at the points
of a do main D a holomorphic function that is sufficiently
well-behaved when approaching the boundary aD, from its values on
aD or on S. Alongside with this classical problem, it is possible
and natural to consider the following one: to recover the
holomorphic function in D from its values on some set MeaD not
containing S. Of course, M is to be a set of uniqueness for the
class of holomorphic functions under consideration (for example,
for the functions continuous in D or belonging to the Hardy class
HP(D), p ~ 1).
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