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 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)

References

[a1]	G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1983)