Multiple Sampling and Interpolation in a Space of Polynomials

Abstract

We study sampling and interpolation arrays with multiplicities for the spaces $\mathcal{P}_k$ of holomorphic polynomials of degree at most $k$. We find that the geometric conditions satisfied by these arrays are in accordance with the conditions satisfied by the sampling and interpolating sequences with unbounded multiplicities in the Fock space, which can be seen as a limiting case of the space $\mathcal{P}_k$ as $k$ tends to infinity. In particular, if the multiplicities tend to infinity, there are no arrays which are simultaneously sampling and interpolating.