Bergman-Weil expansion for holomorphic functions
Date
2022Publisher
SpringerSource
Mathematische AnnalenVolume
382Issue
1-2Pages
383-419Google Scholar check
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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.