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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. Hörmander [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
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