# Bergman-Weil representation

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)