Dirichlet approximation and universal Dirichlet series

We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical results of Runge, Mergelyan and Vitushkin. We also strengthen the notion of universal Dirichlet series.

An asymptotic holomorphic boundary problem on arbitrary open sets in Riemann surfaces

Abstract We show that if U is an arbitrary open subset of a Riemann surface and φ an arbitrary continuous function on the boundary ∂ U , then there exists a holomorphic function φ ˜ on U such that, for every p ∈ ∂ U , φ ˜ ( x ) → φ ( p ) , as x → p outside a set of density 0 at p relative to U . These “solutions to a boundary problem” are not unique. In fact they can be required to have interpolating properties and also to assume all complex values near every boundary point. Our result is new even for the unit disc.

Extendability and domains of holomorphy in infinite-dimensional spaces

A Function Algebra Providing New Mergelyan Type Theorems in Several Complex Variables

For compact sets $K\subset \mathbb C^{d}$, we introduce a subalgebra $A_{D}(K)$ of $A(K)$, which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.