On the Projective Description of Weighted (LF)-Spaces of Continuous Functions

We solve the problem of the topological or algebraic description of countable inductive limits of weighted Fréchet spaces of continuous functions on a cone. This problem is investigated for two families of weights defined by positively homogeneous functions. Weights of this form play the important role in Fourier analysis.

PROJECTIVE DESCRIPTION OF WEIGHTED (LF)-SPACES OF HOLOMORPHIC FUNCTIONS ON THE DISC

The topology of certain weighted inductive limits of Fréchet spaces of holomorphic functions on the unit disc can be described by means of weighted sup-seminorms in case the weights are radial and satisfy certain natural assumptions due to Lusky; in the sense of Shields and Williams the weights have to be normal. It turns out that no assumption on the (double) sequence of normal weights is necessary for the topological projective description in the case of o-growth conditions. For O-growth conditions, we give a necessary and sufficient condition (in terms of associated weights) for projective description in the case of (LB)-spaces and normal weights. This last result is related to a theorem of Mattila, Saksman and Taskinen.AMS 2000 Mathematics subject classification: Primary 46E10. Secondary 30H05; 46A13; 46M40

A PROJECTIVE DESCRIPTION OF THE SPACE OF HOLOMORPHIC GERMS

A natural topology on the set of germs of holomorphic functions on a compact subset $K$ of a Fréchet space is the locally convex inductive limit topology of the spaces $\mathcal{O}(\sOm)$ endowed with the compact open topology; here $\sOm$ is any open subset containing $K$. Mujica gave a description of this space as the inductive limit of a suitable sequence of compact subsets. He used a set of intricate semi-norms for this. We give a projective characterization of this space, using simpler semi-norms, whose form is similar to the one used in the Whitney Extension Theorem for $C_\infty$ functions. They are quite natural in a framework where extensions are involved. We also give a simple proof that this topology is strictly stronger than the topology of the projective limit of the non-quasi-analytic spaces.AMS 2000 Mathematics subject classification: Primary 46A13; 46F15

The locally convex topology on the space of meromorphic functions

We give a positive answer to a question of Horst Tietz. A theorem of his that is related to the Mittag-Leffler theorem looks like a duality restult under some locally convex topology on the space of meromorphic functions. Tietz has posed the problem of finding such a topology. It is shown that a topology introduced by Holdguün in 1973 solves the problem. The main tool in the study of this topology is a projective description of it that is derived here. We also argue that Holdgrün's topology is the natural locally convex topology on the space of meromorphic functions.