Using Bochner-Martinelli type residual currents we prove some generalizations of Jacobi's Residue Formula, which allow proper polynomial maps to have 'common zeroes at infinity ', in projective or toric situations. * AMS classification number: 32A27, (32A25, 32C30).
Carleman formulas, unlike the Cauchy formula, restore a function holomorphic in a domain 𝒟 by its values on a part M of the boundary ∂𝒟, provided that M is of positive Lebesgue measure. An extensive survey of Carleman formulas is found in [AIZ]. In the present paper new Carleman formulas are obtained for domains in ℂn and the question about a description of the class of holomorphic functions that are represented by Carleman formula is investigated.
In [AIZTV] we considered the simplest Carleman formulas in one and several complex variables on particular simply connected domains. It was shown there that a necessary and sufficient condition for a holomorphic function f to be represented by Carleman formula over the set M is that f must belong to “the Hardy class ℋ︁1 near the set M”.
In the present paper we look at the case of Fok‐Kuni integral representation formula. This is a particular form of abstract Carleman formula, but it involves simply connected domains, not covered by previous results. Furthermore we initiate the study of the same questions for non‐simply connected domains. We obtain the description of the class of functions representable by Carleman formula on annulii in ℂ and their generalizations, the Reinhardt domains in ℂn.
Let W be a q-dimensional irreducible algebraic subvariety in the affine space A n C ; P 1 ; y; P m m elements in C½X 1 ; y; X n ; and V ðPÞ the set of common zeros of the P j 's in C n : Assuming that jW j is not included in V ðPÞ; one can attach to P a family of nontrivial W -restricted residual currents in 0 D 0;k ðC n Þ; 1pkpminðm; qÞ; with support on jW j: These currents (constructed following an analytic approach) inherit most of the properties that are fulfilled in the case q ¼ n: When the set jW j-VðPÞ is discrete and m ¼ q; we prove that for every point aAjW j-VðPÞ the W -restricted analytic residue of a ðq; 0Þ-form R dz I ; RAC½X 1 ; y; X n ; at the point a is the same as the residue on W (completion of W in Proj C½X 0 ; y; X n ) at the point a in the sense of Serre (q ¼ 1) or Kunz-Lipman (1oqon) of the q-differential form (R=P 1 ?P q Þdz I : We will present a restricted affine version of Jacobi's residue formula and applications of this formula to higher dimensional analogues of Reiss (or Wood) relations, corresponding to situations where the Zariski closures of jW j and VðPÞ intersect at infinity in an arbitrary way. r
A necessary and sufficient geometric characterization and a necessary and sufficient analytic characterization of interpolating varieties for the space of entire functions will be obtained in the paper, which as an application will also give a generalization of the well-known Pólya-Levinson density theorem.
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