View source for Bergman-Weil representation

''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:

<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>

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

<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>

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

<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>

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,

<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>

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.

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" />.

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

<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

<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>

<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>

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" />.

====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>



====Comments====


====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>