Bernstein Spaces on Siegel CR Manifolds

TL;DR

This paper introduces Bernstein spaces on Siegel CR manifolds, establishing fundamental inequalities and sampling conditions for entire functions of exponential type restricted to these manifolds.

Contribution

It defines Bernstein spaces on a new class of quadratic CR manifolds called Siegel CR manifolds and proves key inequalities and sampling criteria for these spaces.

Findings

Proved the Plancherel-Pólya inequality for these spaces.

Established a Bernstein inequality on Siegel CR manifolds.

Provided a sufficient condition for sequences to be sampling.

Abstract

In this paper we introduce and study Bernstein spaces on a class of quadratic CR manifolds, that we call Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to a given Siegel CR submanifold are $L^p$-integrable with respect to a natural measure. For these spaces, among other results, we prove the Plancherel-P\'olya inequality, a Bernstein inequality and a sufficient condition for a sequence to be sampling.