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 \(\mathbb{C}\).

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 \(\mathbb{C}\).

Reviewer: T.V.Tonev (Missoula)

##### MSC:

30G35 | Functions of hypercomplex variables and generalized variables |

30D30 | Meromorphic functions of one complex variable (general theory) |