Using a modified Cauchy–Weil representation formula in a Weil polyhedron $${\varvec{D}}_f\subset U\subset \mathbb {C}^n$$ D f ⊂ U ⊂ C n , we prove a generalized version of Lagrange interpolation formula (at any order) with respect to a discrete set defined by $$V_{{\varvec{D}}_f}(f):=\{f_1=\cdots = f_m =0\}\, \cap {\varvec{D}}_f$$ V D f ( f ) : = { f 1 = ⋯ = f m = 0 } ∩ D f , when $$m>n$$ m > n and $$\{f_1,\ldots ,f_m\}$$ { f 1 , … , f m } is minimal as a defining system. Thus the set $$V_{{\varvec{D}}_f}(f)$$ V 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 $$m=n$$ 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 $$f^{-1}(\{0\})$$ f - 1 ( { 0 } ) -Lagrange interpolators (at any order) subordinate to the choice of a smooth hermitian metric on the trivial m -bundle $$\mathbb {C}_U^m=U\times \mathbb {C}^m$$ C U m = U × C m , while the mapping $$f = (f_1,\ldots ,f_m)$$ f = ( f 1 , … , f m ) is considered as its section.

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全纯函数的 Bergman-Weil 展开

在 Weil 多面体 $${\varvec{D}}_f\subset U\subset \mathbb {C}^n$$ D f ⊂ U ⊂ C n 中使用改进的 Cauchy-Weil 表示公式，我们证明了关于由 $$V_{{\varvec{D}}_f}(f):=\{f_1=\cdots = f_m =0\}\, \cap 定义的离散集的拉格朗日插值公式（以任意顺序） {\varvec{D}}_f$$ VD f ( f ) : = { f 1 = ⋯ = fm = 0 } ∩ D f ，当 $$m>n$$m > n 且 $$\{f_1,\ ldots ,f_m\}$$ { f 1 , ... , fm } 作为一个定义系统是最小的。因此集合 $$V_{{\varvec{D}}_f}(f)$$ VD f ( f ) 不是完全交集。在 $$m=n$$ m = n 的情况下，我们将我们的结果呈现为经典拉格朗日插值公式的平均版本。为此，我们调用了克罗夫顿公式，它在安德森等人提出的 Vogel 广义循环的构建中起着关键作用。（J Reine Angew Math 728：105–136，2017；Math Ann，2020。https://doi.org/10.1007/s00208-020-01973-y）。这自然地引导我们构建 Bochner-Martinelli 核。我们还引入了 $$f^{-1}(\{0\})$$ f - 1 ( { 0 } ) - 拉格朗日插值器（以任何顺序）从属于在平凡 m 上选择平滑厄密度量 - bundle $$\mathbb {C}_U^m=U\times \mathbb {C}^m$$ CU m = U × C m ，而映射 $$f = (f_1,\ldots ,f_m)$$ f = ( f 1 , … , fm ) 被视为其部分。