Interpolation and sampling arrays in spaces of polynomials

Abstract

We study the discretisation procedure of homogeneous polynomials in the unit sphere $\mathbb{S}\cong \mathbb{CP}^1$. This can be seen as a basic model of a more general problem of discretisation of sections of holomorphic line bundles over compact complex manifolds. Our aim is to obtain geometric necessary and sufficient conditions describing the discretising sequences. An important model for such sequences are the so-called Fekete arrays, which can be seen as nets adapted to the geometry of the sphere. The tools used in such description go back to the signal processing theory pioneered by Beurling and Landau.