Bergman-Weil expansion for holomorphic functions

Abstract

Using a modified Cauchy–Weil representation formula in a Weil polyhedron D f ⊂ U ⊂ Cn, we prove a generalized version of Lagrange interpolation formula (at any order) with respect to a discrete set defined by VD f ( f ) := { f1 =···= fm = 0} ∩ D f , when m > n and { f1,..., fm} is minimal as a defining system. Thus the set VD 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 trivialm-bundleCm U = U×Cm, while the mapping f = ( f1,..., fm)is considered as its section.