A remark on the means of entire Dirichlet series
Let be a class of entire functions represented by Dirichlet series with complex frequencies for which is bounded. Then is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. 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 have also been established.
For sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of the best -approximation with an unbounded number of finite poles are considered.