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Generalized meromorphic functions. (English. Russian original) Zbl 0801.30039
Russ. Acad. Sci., Izv., Math. 42, No. 1, 133-147 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 1, 147-166 (1993).
The autor continues his pioneering work on generalized meromorphic functions on the big plane generated by a compact Abelian group $$G$$ with ordered dual group $$\Gamma\subset\mathbb{R}$$. Here he presents the proofs of several of his previously announced results. Let $$G$$ be a compact Abelian group with ordered dual group $$\Gamma\subset \mathbb{R}$$. The big plane over $$G$$ is the infinite cone $$\mathbb{C}_ \Gamma= [0,\infty)\cdot G$$, the unit big disc $$\Omega$$ over $$G$$ is the set of points $${\mathbf w}= rg$$ in the big plane whose “modulus” $$|{\mathbf w}|= r$$ is $$\leq 1$$, and $$\Omega^ 0$$ is its interior. A continuous function $$f$$ in a domain $$D\subset \mathbb{C}_ \Gamma$$ is “analytic” (generalized analytic) in $$D$$ if $$f$$ can be approximated locally by linear combinations $$\sum c(a)\backslash f(a)$$ over $$\mathbb{C}$$ of functions $$f^ a(rg)= r^ a g(a)$$, where $$r\geq 0,$$, $$g\in G$$ and $$a\in \Gamma_ +=\Gamma\cap [0,\infty)$$ in $$\mathbb{C}_ \Gamma$$.
For $$D=\Omega$$ the analyticity in this context was introduced by R. Arens and I. Singer in 1956; for an arbitrary $$D$$ the notion is due to D. Stankov and the reviewer [e.g., Big planes, boundaries and function algebras (1992; Zbl 0755.46020)]. We mention only a few of the many results in this paper.
It is given a description of the measures on $$G$$ that are orthogonal to the disc algebra of continuous up to the boundary $$G$$ generalized analytic functions on $$\Omega$$. The proof of author’s result for unique generalized analytic extension on a domain $$D\subset\Omega^ 0$$ of a bounded generalized analytic function defined on the complement in $$D$$ of a certain thin set in $$\mathbb{C}_ \Gamma$$ is presented as well.
For a class of suitably defined meromorphic functions in $$\mathbb{C}_ \Gamma$$ the following factorization result is proved.
Theorem. Let $$f$$ be a meromorphic function in $$\Omega^ 0$$ and let $$S^*\in \Omega^ 0$$ is either a removable singularity or an isolated pole. Then there is a non-vanishing generalized analytic function $$g$$ on $$\Omega^ 0$$, such that $$f\cdot g$$ can be extended to a generalized analytic function on $$\Omega^ 0$$. Versions of this result are given for the case of meromorphic functions on a “big annulus” region of type $$E^ 0=\{{\mathbf w}\in \mathbb{C}_ \Gamma< r|{\mathbf w}|< 1\}$$ and for meromorphic almost periodic functions on the upper half plane or on a horizontal strip in $$\mathbb{C}$$.

##### MSC:
 30G35 Functions of hypercomplex variables and generalized variables 30D30 Meromorphic functions of one complex variable (general theory)
big plane
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