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On generators in some rings of analytic functions. (English. Russian original) Zbl 0798.32016
Russ. Acad. Sci., Dokl., Math. 46, No. 1, 131-134 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 325, No. 5, 919-922 (1992).
Let $${\mathcal D}$$ be a domain in $$\mathbb{C}^ n$$ and let $$\psi : \mathbb{C}^ n \to R^ +$$ be a plurisubharmonic function satisfying some mild additional conditions. Let $$H_ \psi ({\mathcal D})$$ be the set of all analytic functions $$f$$ on $${\mathcal D}$$ satisfying the growth condition $$| f(z) | \leq a_ 1 \exp \{a_ 2 \psi (z)\}$$ for all $$z \in {\mathcal D}$$, where the constants $$a_ 1$$ and $$a_ 2$$ depend on $$f$$. If $$p>0$$, let $${\mathcal H}^ p_ \psi$$ be the set of sequences $$(f_ j)$$ of elements of $$H_ \psi ({\mathcal D})$$ such that $$\sum^ \infty_{j = 1} | f_ j (z) |^ p \leq a_ 1 \exp \{a_ 2 \psi (z)\}$$ for all $$z\in {\mathcal D}$$. For a fixed sequence $$f=(f_ j) \in {\mathcal H}^ p_ \psi$$ let $$I(f) = \{\varphi \in H_ \psi ({\mathcal D}):$$ $$\varphi = \sum^ \infty_{j=1} f_ j g_ j\}$$, where the sequence $$(g_ j)$$ belongs to the conjugate space $${\mathcal H}^ s_ \psi$$.
The authors investigate when $$I(f) = H_ \psi ({\mathcal D})$$, thereby extending their own as well as work of L. HĂ¶rmander [Bull. Am. Math. Soc. 73, 943-949 (1967; Zbl 0172.417)], J. Kelleher and B. A. Taylor [Bull. Am. Math. Soc. 73, 246-249 (1967; Zbl 0154.150)], and others.
Reviewer: R.M.Aron (Kent)

##### MSC:
 32A38 Algebras of holomorphic functions of several complex variables 32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs 32U05 Plurisubharmonic functions and generalizations 46J20 Ideals, maximal ideals, boundaries