Mathematics – Complex Variables
Scientific paper
2011-10-26
Mathematics
Complex Variables
This is a preliminary version, with a number of results not included. The complete version will appear shortly
Scientific paper
We develop the elements of complex function theory within certain algebras of holomorphic functions on coverings of complex manifolds (including holomorphic extension from complex submanifolds, properties of divisors, corona type theorem, holomorphic analogue of Peter-Weyl approximation theorem, Hartogs type theorem, characterization of the uniqueness sets, etc). Our model examples are: (1) algebra of Bohr's holomorphic almost periodic functions on tube domains (i.e. the uniform limits of exponential polynomials) (2) algebra of all fibrewise bounded holomorphic functions (arising in corona problem for H^\infty) (3) algebra of holomorphic functions having fibrewise limits. Our proofs are based on the analogues of Cartan theorems A and B for coherent type sheaves on the maximal ideal spaces of these subalgebras.
Brudnyi Alexander
Kinzebulatov Damir
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