Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables. (English) Zbl 1180.32002
Let \(\mathcal{F}(R)\) be the set of holomorphic functions on a Reinhardt domain \(R\) in a Banach sequence space. If \(f\in\mathcal{F}(R)\), then \(f\) has a monomial expansion of the form \(\sum_{\alpha}\frac{\partial^{\alpha}f(0)}{\alpha!}z^{\alpha}\).
Denote by \(\mathrm{dom}\mathcal{F}(R)\) the domain of convergence of \(\mathcal{F}(R)\), that is the set of all elements \(z\in R\) for which the monomial expansion of each function \(f\in\mathcal{F}(R)\) converges.
In the paper, the authors give a systematic study of the sets \(\mathrm{dom}\mathcal{F}(R)\).
Denote by \(\mathrm{dom}\mathcal{F}(R)\) the domain of convergence of \(\mathcal{F}(R)\), that is the set of all elements \(z\in R\) for which the monomial expansion of each function \(f\in\mathcal{F}(R)\) converges.
In the paper, the authors give a systematic study of the sets \(\mathrm{dom}\mathcal{F}(R)\).
Reviewer: Dorina Raducanu (Brasov)
MSC:
| 32A10 | Holomorphic functions of several complex variables |
| 32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |
Keywords:
holomorphic function; Reinhardt domain; Banach sequence space; monomial expansion; domain of convergenceReferences:
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