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The divisor of a generalized analytic function. (English. Russian original) Zbl 0914.30038
Math. Notes 61, No. 5, 547-552 (1997); translation from Mat. Zametki 61, No. 5, 655-661 (1997).
Let \({\mathcal O}(\Delta_G)\) be the space of generalized analytic functions in the open big disc \(\Delta_G\), \(\widehat G\subset {\mathbb R},\) in the sense of Arens and Singer. The spectrum Sp \(f\) of \(f\) is the set of all \(a\in \widehat G\cap [0,\infty)\) with non-zero Fourier coefficients \(c_a=\int_G\frac{f(rg)}{\chi^a(rg)} d\sigma(g)\), where \((rg)=s\in\Delta_G\), \(g\in G\), \(0\leq r<1\), \(\chi^a\in\widehat G\), and \(\sigma\) is the normalized Haar measure on \(G\). It is well known that for any given \(s\in\Delta_G\) there is an embedding of the upper half plane into \(\Delta_G\) through \(s\) and the restrictions \(\widetilde f\) of generalized analytic functions \(f\) on its image are analytic (and almost periodic) on the upper half-plane. The author introduces the notion of divisor of a generalized analytic function. Namely, \(\text{div,} f(s)= \text{ord } \widetilde f(s)\) for \(s\in\Delta_G\setminus \{O\}\), and, \(\text{div }f(O) = \inf\{\text{Sp } f\}\). A function \(g \in{\mathcal O}(\Delta_G)\) is said to divide \(f\in {\mathcal O}(\Delta_G)\) if \(f=gh\) for some \(h\in{\mathcal O}(\Delta_G)\).
Main theorem: If \(f,g\in {\mathcal O}(\Delta_G)\), then \(g\) divides \(f\) if and only if \(\text{div }g\leq \text{div } f\). As a consequence, given \(\text{div } f = \text{div } g\) one can find an \(h\in{\mathcal O}(\Delta_G)\) such that \(f=g\exp h\).
30G35 Functions of hypercomplex variables and generalized variables
Full Text: DOI
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