Bergman–Weil expansion for holomorphic functions

Abstract

Using a modified Cauchy–Weil representation formula in a Weil polyhedron ${\varvec{D}}_f\subset U\subset \mathbb {C}^n$, we prove a generalized version of Lagrange interpolation formula (at any order) with respect to a discrete set defined by $V_{{\varvec{D}}_f}(f):=\{f_1=\cdots = f_m =0\}\, \cap {\varvec{D}}_f$, when $m>n$ and $\{f_1,\ldots ,f_m\}$ is minimal as a defining system. Thus the set $V_{{\varvec{D}}_f}(f)$ fails to be a complete intersection. We present our result as an averaged version of the classic Lagrange interpolation formula in the case $m=n$. We invoke to that purpose Crofton’s formula, which plays a key role in the construction of Vogel generalized cycles as proposed in Andersson et al. (J Reine Angew Math 728: 105–136, 2017; Math Ann, 2020. https://doi.org/10.1007/s00208-020-01973-y). This leads us naturally to the construction of Bochner–Martinelli kernels. We also introduce $f^{-1}(\{0\})$-Lagrange interpolators (at any order) subordinate to the choice of a smooth hermitian metric on the trivial m-bundle $\mathbb {C}_U^m=U\times \mathbb {C}^m$, while the mapping $f = (f_1,\ldots ,f_m)$ is considered as its section.