**"Sampling and interpolation sets in several complex variables"**
#### Haimi, AnttiGiven a plurisubharmonic function $\phi$ on $\mathbb{C}^n$, we
consider the space of entire functions that are square integrable with
respect to the weight $e^{-\phi}$. A canonical special case is the
Bargmann-Fock space which corresponds to $\phi(z)= |z|^2$. We are
interested in density of sampling and interpolation sets in these spaces.
In one variable there is a characterization in terms of densities but in
several variables this is not possible. However, using a method going back
to Landau, necessary density conditions can be given. Resolving a
conjecture of Lindholm, we show that these density conditions are strict,
i.e. that there are no sampling or interpolation sets on the critical
density. Our method is based on the following ingredients: Beurling's weak
limit technique, Sjöstrand's Wiener-type lemma, translation-type operators
introduced by Ortega-Cerda and Seip in the one variable context.
Joint work with Gröchenig, Ortega-Cerda and Romero |