2016, Aron, Richard M., Falcó, Javier, García, Domingo, Maestre, Manuel

Let BX be the open unit ball of a complex Banach space X, and let H∞(BX) and Au(BX) be, respectively, the algebra of bounded holomorphic functions on BX and the subalgebra of uniformly continuous holomorphic functions on BX. In this paper we study the analytic structure of fibers in the spectrum of these two algebras. For the case of H∞(BX), we prove that the fiber in M(H∞(Bc0)) over any point of the distinguished boundary of the closed unit ball B¯ℓ∞ of ℓ∞ contains an analytic copy of Bℓ∞. In the case of Au(BX) we prove that if there exists a polynomial whose restriction to the open unit ball of X is not weakly continuous at some point, then the fiber over every point of the open unit ball of the bidual contains an analytic copy of D.

2016, Aron, Richard M., Bayart, Frédéric, Gauthier, Paul M., Maestre, Manuel, Nestoridis, Vassili

We characterize the uniform limits of Dirichlet polynomials on a right half plane. We extend the approximation theorems of Runge,Mergelyan and Vitushkin to the Dirichlet setting with respect to the Euclidean distance and to the chordal one, as well. We also strengthen the notion of Universal Dirichlet series.

2016, Aron, Richard M., Gauthier, Paul M., Maestre, Manuel, Nestoridis, Vassili, Falcó, Javier

We consider A(Ω), the Banach space of functions f from Ω¯¯¯¯=∏i∈IUi¯¯¯¯¯ to C that are continuous with respect to the product topology and separately holomorphic, where I is an arbitrary set and Ui are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off Ui¯¯¯¯¯ are uniformly dense in A(Ω). This generalizes previous results where Ui=D is the open unit disc in C or Ui¯¯¯¯¯c is connected.