Multiple sampling and interpolation in a space of polynomials

TL;DR

This paper investigates sampling and interpolation arrays with multiplicities in polynomial spaces, revealing their geometric conditions align with those in the Fock space as the degree grows large.

Contribution

It establishes a connection between polynomial space sampling/interpolation arrays and Fock space sequences, extending understanding of their geometric conditions.

Findings

Geometric conditions for polynomial arrays match those in Fock space.

Sampling and interpolation arrays with multiplicities are characterized.

Limit behavior as polynomial degree tends to infinity is analyzed.

Abstract

We study sampling and interpolation arrays with multiplicities for the spaces 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 P_k as k tends to infinity.