Multiple Sampling and Interpolation in the Classical Fock Space

Abstract

We study multiple sampling, interpolation and uni- queness for the classical Fock space in the case of unbounded mul- tiplicities. Sampling and interpolating sequences in Fock spaces were character- ized by Seip and Seip-Wallsten in (3, 4) by means of a certain Beurling- type asymptotic uniform density. The case of uniformly bounded mul- tiplicities was considered by Brekke and Seip (1) who gave a complete description in this situation. Their conditions show that it is not pos- sible that a sequence is simultaneously sampling and interpolating. Brekke and Seip also asked whether there exist sequences which are simultaneously sampling and interpolating when the multiplicities are unbounded. In this research note we formulate some conditions (of geometric nature) for sampling and interpolation. They show that the answer to this question is negative when the multiplicities tend to infinity. The detailed version of this work will be published elsewhere.