Sampling in spaces of entire functions of exponential type in $\mathbb C^{n+1}$

TL;DR

This paper develops sampling theorems for entire functions of exponential type in several variables, with growth conditions related to the boundary of the Siegel upper half-space, leveraging the structure of the Heisenberg group.

Contribution

It introduces a new growth condition for entire functions in multiple variables and establishes sampling theorems using harmonic analysis on the Heisenberg group.

Findings

Proves a Whittaker–Kotelnikov–Shannon type sampling theorem.

Establishes a Paley–Wiener type theorem for these function spaces.

Derives a Plancherel–Pólya inequality for functions restricted to the hypersurface.

Abstract

In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary $b\mathcal U$ of the Siegel upper half-space $\mathcal U$ and it is fundamental that $b\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. We consider entire functions in $\mathbb C^{n+1}$ of exponential type with respect to the hypersurface $b\mathcal U$ whose restriction to $b\mathcal U$ are square integrable with respect to the Haar measure on $\mathbb H_n$. For these functions we prove a version of the Whittaker--Kotelnikov--Shannon Theorem. Instrumental in our work are spaces of entire functions in $\mathbb C^{n+1}$ of exponential type with respect to the hypersurface $b\mathcal U$ whose restrictions to $b\mathcal U$ belong to some homogeneous Sobolev space on $\mathbb H_n$. For these spaces, using the group Fourier transform on $\mathbb H_n$, we prove a Paley--Wiener type theorem and a Plancherel--P\'olya type inequality.