Harmonic Analysis and Applications

June 4-8, 2018


"Sampling and interpolation sets in several complex variables"

Haimi, Antti

Given 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

« back