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Bergman–Weil formula, Weil formula
An integral representation of holomorphic functions, obtained by S. Bergman [1] and A. Weil [2] and defined as follows. Let be a domain of holomorphy in
, let the functions
be holomorphic in
and let
compactly belong to
. It is then possible to represent any function
holomorphic in
and continuous on
at any point
by the formula:
![]() | (*) |
where the summation is performed over all , while the integration is carried out over suitably-oriented
-dimensional surfaces
, forming the skeleton of the domain
(cf. Analytic polyhedron),
. Here the functions
are holomorphic in the domain
and are defined, in accordance with Hefer's lemma [3], by the equations
![]() |
The integral representation (*) is called the Bergman–Weil representation.
The domains appearing in the Bergman–Weil representation are called Weil domains; an additional condition must usually be imposed, viz. that the ranks of the matrices
,
,
,
, on the corresponding sets
![]() |
are maximal for all
(such Weil domains are called regular). The Weil domains in the Bergman–Weil representations may be replaced by analytic polyhedra
compactly belonging to D,
![]() |
where the are bounded domains with piecewise-smooth boundaries
in the plane
. The Bergman–Weil representation defines the value of a holomorphic function
inside the analytic polyhedron
from the values of
on the skeleton
; for
the dimension of
is strictly lower than that of
. If
, analytic polyhedra become degenerate in a domain with piecewise-smooth boundary, the skeleton and the boundary become identical, and if, moreover,
and
, then the Bergman–Weil representation becomes identical with Cauchy's integral formula.
An important property of the Bergman–Weil representation is that its kernel is holomorphic in . Accordingly, if the holomorphic function
is replaced by an arbitrary function which is integrable over
, then the right-hand side of the Weil representation gives a function which is holomorphic everywhere in
and almost-everywhere in
; such functions are called integrals of Bergman–Weil type. If
is holomorphic in
and continuous on
, then its integral of Bergman–Weil type is zero almost-everywhere on
.
Bergman–Weil representations in a Weil domain yield, after the substitution
![]() |
the Weil decomposition
![]() |
![]() |
into a series of functions, holomorphic in , and this series is uniformly convergent on compact subsets of
.
References
[1] | S.B. Bergman, Mat. Sb. , 1 (43) (1936) pp. 242–257 |
[2] | A. Weil, "L'intégrale de Cauchy et les fonctions de plusieurs variables" Math. Ann. , 111 (1935) pp. 178–182 |
[3] | V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian) |
Comments
References
[a1] | G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1983) |
Bergman-Weil representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bergman-Weil_representation&oldid=22095