Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables
Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables
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DOI：
10.1006/jmaa.2000.7355
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发表时间：
2001-06
影响因子：
1.3
通讯作者：
L. Aĭzenberg;A. Aytuna;P. Djakov
中科院分区：
文献类型：
--
作者：
L. Aĭzenberg;A. Aytuna;P. Djakov
文摘在第一部分,我们推广了经典结果证明波尔的一个类似的现象发生时D是一个开放的领域在C m(或者,更普遍的是,一个复杂的多方面的)和(ϕn)∞n = 0是一个基础的空间全纯函数的H (D)使得ϕ0 = 1,ϕn (z0) = 0, n≥1,对于一些z0∈D .也就是说,存在一个邻居U的点z0,每当上全纯函数D模数小于1,U上项的模的展开式的和也小于1。第二部分考虑了解析函数的一些自然希尔伯特空间，并推导了在这种情况下玻尔现象发生的充分必要条件。
Abstract In the first part, we generalize the classical result of Bohr by proving that an analogous phenomenon occurs whenever D is an open domain in C m (or, more generally, a complex manifold) and (ϕn)∞n = 0 is a basis in the space of holomorphic functions H(D) such that ϕ0 = 1 and ϕn(z0) = 0, n ≥ 1, for some z0 ∈ D. Namely, then there exists a neighborhood U of the point z0 such that, whenever a holomorphic function on D has modulus less than 1, the sum of the suprema in U of the moduli of the terms of its expansion is less than 1 too. In the second part we consider some natural Hilbert spaces of analytic functions and derive necessary and sufficient conditions for the occurrence of Bohr's phenomenon in this setting.