Analysis Seminar

Let Y denote the space of holomorphic functions in a planar
domain Omega, such that all order's derivatives extend continuously on the closure of
Omega in the plane C. We endow Y with its natural topology and let X
denote the closure in Y of all rational functions with poles off the
closure of Omega. Some universality results concerning Taylor series or
Pade approximants are generic in X. In order to strengthen the above
results we give a sufficient condition of geometric nature assuring that
X=Y. In addition to this, if a Jordan domain Omega satisfies the above
condition, then the primitive F of a holomorphic function f in Omega is at
least as smooth on the boundary as f, even if the boundary of Omega has
infinite length. This led us to construct a Jordan domain Omega supporting
a holomorphic function f which extends continuously on the closure of
Omega, such that its primitive F is even not bounded in Omega. Finally we
extend the last result in generic form to more general Volterra operators.
(This is based on a joint work with Ilias Zadik.)