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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 $$f=g\exp h$$.
MSC:
 30G35 Functions of hypercomplex variables and generalized variables
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References:
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