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''Bergman–Weil formula, Weil formula''
 
''Bergman–Weil formula, Weil formula''
  
An integral representation of holomorphic functions, obtained by S. Bergman [[#References|[1]]] and A. Weil [[#References|[2]]] and defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155701.png" /> be a domain of holomorphy in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155702.png" />, let the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155703.png" /> be holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155704.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155705.png" /> compactly belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155706.png" />. It is then possible to represent any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155707.png" /> holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155708.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b0155709.png" /> at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557010.png" /> by the formula:
+
An integral representation of holomorphic functions, obtained by S. Bergman [[#References|[1]]] and A. Weil [[#References|[2]]] and defined as follows. Let $  D $
 +
be a domain of holomorphy in $  \mathbf C  ^ {n} $,  
 +
let the functions $  W _ {1} \dots W _ {j} $
 +
be holomorphic in $  D $
 +
and let  $  V = \{ {z \in D } : {| W _ {k} (z) | < 1,  k = 1 \dots N } \} $
 +
compactly belong to $  D $.  
 +
It is then possible to represent any function $  f $
 +
holomorphic in $  V $
 +
and continuous on $  \overline{V}\; $
 +
at any point $  z \in V $
 +
by the formula:
 +
 
 +
$$ \tag{* }
 +
f(z)  =  {
 +
\frac{1}{(2 \pi i)  ^ {n} }
 +
}
 +
\sum \int\limits
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\frac{f ( \zeta )  \mathop{\rm det}  (P _ {ij _ {k}  } ) }{\prod _ { k=1 } ^ { n }
 +
(W _ {j _ {k}  } ( \zeta ) - W _ {j _ {k}  } (z)) }
 +
\
 +
d \zeta ,
 +
$$
  
where the summation is performed over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557012.png" />, while the integration is carried out over suitably-oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557013.png" />-dimensional surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557014.png" />, forming the skeleton of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557015.png" /> (cf. [[Analytic polyhedron|Analytic polyhedron]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557016.png" />. Here the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557017.png" /> are holomorphic in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557018.png" /> and are defined, in accordance with Hefer's lemma [[#References|[3]]], by the equations
+
where the summation is performed over all $  j _ {1} < \dots < j _ {n} $,  
 +
while the integration is carried out over suitably-oriented $  n $-
 +
dimensional surfaces $  \sigma _ {j _ {1}  \dots j _ {n} } $,  
 +
forming the skeleton of the domain $  V $(
 +
cf. [[Analytic polyhedron|Analytic polyhedron]]), $  d \zeta = d \zeta _ {1} \wedge \dots \wedge d \zeta _ {n} $.  
 +
Here the functions $  P _ {ij }  ( \zeta , z) $
 +
are holomorphic in the domain $  D \times D $
 +
and are defined, in accordance with Hefer's lemma [[#References|[3]]], by the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557019.png" /></td> </tr></table>
+
$$
 +
W _ {j} ( \zeta ) - W _ {j} (z)  = \
 +
\sum _ { i=1 } ^ { n }
 +
( \zeta _ {i} - z _ {i} ) P _ {ij} ( \zeta , z).
 +
$$
  
 
The integral representation (*) is called the Bergman–Weil representation.
 
The integral representation (*) is called the Bergman–Weil representation.
  
The domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557020.png" /> appearing in the Bergman–Weil representation are called Weil domains; an additional condition must usually be imposed, viz. that the ranks of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557024.png" />, on the corresponding sets
+
The domains $  V $
 +
appearing in the Bergman–Weil representation are called Weil domains; an additional condition must usually be imposed, viz. that the ranks of the matrices $  ( \partial  W _ {j _  \nu  } / \partial  z _  \mu  ) $,  
 +
$  \nu = 1 \dots k $,
 +
$  \mu = 1 \dots n $,
 +
$  k \leq  n $,
 +
on the corresponding sets
 +
 
 +
$$
 +
\{ {z \in \overline{V}\; } : {| W _ {j _ {1}  } |
 +
= \dots = | W _ {j _ {k}  } | = 1 } \}
 +
$$
 +
 
 +
are maximal  $  (=k) $
 +
for all  $  j _ {1} < \dots < j _ {k} $(
 +
such Weil domains are called regular). The Weil domains in the Bergman–Weil representations may be replaced by analytic polyhedra  $  U $
 +
compactly belonging to D,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557025.png" /></td> </tr></table>
+
$$
 +
= \{ {z \in D } : {W _ {j} (z) \in D _ {j} ,\
 +
j =1 \dots N } \}
 +
,
 +
$$
  
are maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557027.png" /> (such Weil domains are called regular). The Weil domains in the Bergman–Weil representations may be replaced by analytic polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557028.png" /> compactly belonging to D,
+
where the  $  D _ {j} $
 +
are bounded domains with piecewise-smooth boundaries  $  \partial  D _ {j} $
 +
in the plane  $  \mathbf C $.  
 +
The Bergman–Weil representation defines the value of a holomorphic function  $  f $
 +
inside the analytic polyhedron  $  U $
 +
from the values of  $  f $
 +
on the skeleton  $  \sigma $;
 +
for  $  n > 1 $
 +
the dimension of  $  \sigma $
 +
is strictly lower than that of  $  \partial  U $.  
 +
If  $  n = 1 $,
 +
analytic polyhedra become degenerate in a domain with piecewise-smooth boundary, the skeleton and the boundary become identical, and if, moreover,  $  N = 1 $
 +
and  $  W(z) = z $,
 +
then the Bergman–Weil representation becomes identical with Cauchy's integral formula.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557029.png" /></td> </tr></table>
+
An important property of the Bergman–Weil representation is that its kernel is holomorphic in  $  z $.
 +
Accordingly, if the holomorphic function  $  f $
 +
is replaced by an arbitrary function which is integrable over  $  \sigma $,
 +
then the right-hand side of the Weil representation gives a function which is holomorphic everywhere in  $  U $
 +
and almost-everywhere in  $  D \setminus  \partial  U $;  
 +
such functions are called integrals of Bergman–Weil type. If  $  f $
 +
is holomorphic in  $  U $
 +
and continuous on  $  \overline{U}\; $,
 +
then its integral of Bergman–Weil type is zero almost-everywhere on  $  D \setminus  \overline{U}\; $.
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557030.png" /> are bounded domains with piecewise-smooth boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557031.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557032.png" />. The Bergman–Weil representation defines the value of a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557033.png" /> inside the analytic polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557034.png" /> from the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557035.png" /> on the skeleton <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557036.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557037.png" /> the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557038.png" /> is strictly lower than that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557040.png" />, analytic polyhedra become degenerate in a domain with piecewise-smooth boundary, the skeleton and the boundary become identical, and if, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557042.png" />, then the Bergman–Weil representation becomes identical with Cauchy's integral formula.
+
Bergman–Weil representations in a Weil domain $  V $
 +
yield, after the substitution
  
An important property of the Bergman–Weil representation is that its kernel is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557043.png" />. Accordingly, if the holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557044.png" /> is replaced by an arbitrary function which is integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557045.png" />, then the right-hand side of the Weil representation gives a function which is holomorphic everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557046.png" /> and almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557047.png" />; such functions are called integrals of Bergman–Weil type. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557048.png" /> is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557049.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557050.png" />, then its integral of Bergman–Weil type is zero almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557051.png" />.
+
$$
 +
(W _ {j _ {k}  } ( \zeta ) - W _ {j _ {k}  } (z))  ^ {-1}  = \
 +
\sum _ { v=0 } ^  \infty 
  
Bergman–Weil representations in a Weil domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557052.png" /> yield, after the substitution
+
\frac{W _ {j _ {k}  } ^ { v } (z) }{W _ {j _ {k}  } ^ { v+1 } ( \zeta ) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557053.png" /></td> </tr></table>
+
$$
  
 
the Weil decomposition
 
the Weil decomposition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557054.png" /></td> </tr></table>
+
$$
 +
f (z) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557055.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {s _ {k} \geq  0 }  \sum _ {j _ {1} < \dots < j _ {k} }  Q _ {j _ {1}  \dots j _ {n} s _ {1} \dots s _ {n} } (z) (W _ {j _ {1}  } ^ { s _ {1} }
 +
(z) \dots W _ {j _ {k}  } ^ { s _ {k} } (z))
 +
$$
  
into a series of functions, holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557056.png" />, and this series is uniformly convergent on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015570/b01557057.png" />.
+
into a series of functions, holomorphic in $  D $,  
 +
and this series is uniformly convergent on compact subsets of $  V $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.B. Bergman,  ''Mat. Sb.'' , '''1 (43)'''  (1936)  pp. 242–257</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "L'intégrale de Cauchy et les fonctions de plusieurs variables"  ''Math. Ann.'' , '''111'''  (1935)  pp. 178–182</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.B. Bergman,  ''Mat. Sb.'' , '''1 (43)'''  (1936)  pp. 242–257</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "L'intégrale de Cauchy et les fonctions de plusieurs variables"  ''Math. Ann.'' , '''111'''  (1935)  pp. 178–182</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.M. [G.M. Khenkin] Henkin,  J. Leiterer,  "Theory of functions on complex manifolds" , Birkhäuser  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.M. [G.M. Khenkin] Henkin,  J. Leiterer,  "Theory of functions on complex manifolds" , Birkhäuser  (1983)</TD></TR></table>

Latest revision as of 10:58, 29 May 2020


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 $ D $ be a domain of holomorphy in $ \mathbf C ^ {n} $, let the functions $ W _ {1} \dots W _ {j} $ be holomorphic in $ D $ and let $ V = \{ {z \in D } : {| W _ {k} (z) | < 1, k = 1 \dots N } \} $ compactly belong to $ D $. It is then possible to represent any function $ f $ holomorphic in $ V $ and continuous on $ \overline{V}\; $ at any point $ z \in V $ by the formula:

$$ \tag{* } f(z) = { \frac{1}{(2 \pi i) ^ {n} } } \sum \int\limits \frac{f ( \zeta ) \mathop{\rm det} (P _ {ij _ {k} } ) }{\prod _ { k=1 } ^ { n } (W _ {j _ {k} } ( \zeta ) - W _ {j _ {k} } (z)) } \ d \zeta , $$

where the summation is performed over all $ j _ {1} < \dots < j _ {n} $, while the integration is carried out over suitably-oriented $ n $- dimensional surfaces $ \sigma _ {j _ {1} \dots j _ {n} } $, forming the skeleton of the domain $ V $( cf. Analytic polyhedron), $ d \zeta = d \zeta _ {1} \wedge \dots \wedge d \zeta _ {n} $. Here the functions $ P _ {ij } ( \zeta , z) $ are holomorphic in the domain $ D \times D $ and are defined, in accordance with Hefer's lemma [3], by the equations

$$ W _ {j} ( \zeta ) - W _ {j} (z) = \ \sum _ { i=1 } ^ { n } ( \zeta _ {i} - z _ {i} ) P _ {ij} ( \zeta , z). $$

The integral representation (*) is called the Bergman–Weil representation.

The domains $ V $ appearing in the Bergman–Weil representation are called Weil domains; an additional condition must usually be imposed, viz. that the ranks of the matrices $ ( \partial W _ {j _ \nu } / \partial z _ \mu ) $, $ \nu = 1 \dots k $, $ \mu = 1 \dots n $, $ k \leq n $, on the corresponding sets

$$ \{ {z \in \overline{V}\; } : {| W _ {j _ {1} } | = \dots = | W _ {j _ {k} } | = 1 } \} $$

are maximal $ (=k) $ for all $ j _ {1} < \dots < j _ {k} $( such Weil domains are called regular). The Weil domains in the Bergman–Weil representations may be replaced by analytic polyhedra $ U $ compactly belonging to D,

$$ U = \{ {z \in D } : {W _ {j} (z) \in D _ {j} ,\ j =1 \dots N } \} , $$

where the $ D _ {j} $ are bounded domains with piecewise-smooth boundaries $ \partial D _ {j} $ in the plane $ \mathbf C $. The Bergman–Weil representation defines the value of a holomorphic function $ f $ inside the analytic polyhedron $ U $ from the values of $ f $ on the skeleton $ \sigma $; for $ n > 1 $ the dimension of $ \sigma $ is strictly lower than that of $ \partial U $. If $ n = 1 $, analytic polyhedra become degenerate in a domain with piecewise-smooth boundary, the skeleton and the boundary become identical, and if, moreover, $ N = 1 $ and $ W(z) = z $, 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 $ z $. Accordingly, if the holomorphic function $ f $ is replaced by an arbitrary function which is integrable over $ \sigma $, then the right-hand side of the Weil representation gives a function which is holomorphic everywhere in $ U $ and almost-everywhere in $ D \setminus \partial U $; such functions are called integrals of Bergman–Weil type. If $ f $ is holomorphic in $ U $ and continuous on $ \overline{U}\; $, then its integral of Bergman–Weil type is zero almost-everywhere on $ D \setminus \overline{U}\; $.

Bergman–Weil representations in a Weil domain $ V $ yield, after the substitution

$$ (W _ {j _ {k} } ( \zeta ) - W _ {j _ {k} } (z)) ^ {-1} = \ \sum _ { v=0 } ^ \infty \frac{W _ {j _ {k} } ^ { v } (z) }{W _ {j _ {k} } ^ { v+1 } ( \zeta ) } $$

the Weil decomposition

$$ f (z) = $$

$$ = \ \sum _ {s _ {k} \geq 0 } \sum _ {j _ {1} < \dots < j _ {k} } Q _ {j _ {1} \dots j _ {n} s _ {1} \dots s _ {n} } (z) (W _ {j _ {1} } ^ { s _ {1} } (z) \dots W _ {j _ {k} } ^ { s _ {k} } (z)) $$

into a series of functions, holomorphic in $ D $, and this series is uniformly convergent on compact subsets of $ V $.

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)
How to Cite This Entry:
Bergman-Weil representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bergman-Weil_representation&oldid=18015
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article