# Some functions spaces in the slice hyperholomorphic setting

Presenter: Irene Sabadini, Politecnico di Milano, Italy

Joint work with Daniel Alpay, Vladimir Bolotnikov,  Fabrizio Colombo, J. Oscar Gonzales-Cervantes, Guy Salomon

In this talk we will discuss some function spaces of slice hyperholomorphic functions. In particular, we will treat the case of the Hardy space on the unit ball and on the half space, Blaschke products and applications, like the Krein-Langer factorization. We will also introduce the two formulations of the Bergman theory in this setting and show the relations between them. Finally, if the time permits, we will sketch the notion of Fock space.

# Approximation properties for functions of a quaternionic variable

Presenter: Irene Sabadini, Politecnico di Milano, Italy

Joint work with S. Gal

In this talk we discuss some approximation properties of slice hyperholomorphic functions of one quaternionic variable. As it is well known, this class of functions contains converging power series of the variable. We show that an almost universal property holds: for any $0 < r < R$, there exists a quaternionic power series of radius $r$, with the property that for any axially symmetric open set $\Omega$ with $\Omega \bigcap \overline{B(0;R)}=\emptyset$, any compact axially symmetric $K\subset \Omega$ with $\mathbb{H}\setminus K$ connected, $\mathbb{C}_I\setminus (K\cap \mathbb{C}_I)$ connected (for all $I$ in the sphere of purely imaginary unitary quaternions) and $h:\Omega\to \mathbb{C}$ slice regular in $\Omega$, there exists a subsequence of the partial sums sequence which converges uniformly to $h$ on $K$. We also discuss the validity of Mergelyan-type results, by showing uniform approximation results of slice hyperholomorphic functions by polynomials, in the case of starlike sets and axially symmetric sets.

S. Gal, I. Sabadini, Walsh Equiconvergence Theorems in the Quaternionic Setting, Complex Variables and Elliptic Equations, to appear.

S. Gal, I. Sabadini, Universality properties of the quaternionic power series and entire functions, preprint, 2014.

S. Gal, I. Sabadini, Approximation by polynomials on quaternionic compact sets, preprint, 2013.