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A study of various results for a class of entire Dirichlet series with complex frequencies. (English) Zbl 1463.30011

Summary: Let \(F\) be a class of entire functions represented by Dirichlet series with complex frequencies \(\sum a_k\text{e}^{\langle\lambda ^k,z\rangle}\) for which \((|\lambda ^k|/\text{e})^{|\lambda ^k|}k!|a_k|\) is bounded. Then \(F\) is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. \(F\) is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to \(F\) have also been established.

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
17A35 Nonassociative division algebras
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