Sampling in de Branges Spaces and Naimark Dilation

TL;DR

This paper investigates sampling in de Branges spaces, establishing conditions for sampling sequences through Naimark dilation, which generalizes classical results in Paley-Wiener spaces by embedding into larger spaces with Riesz bases.

Contribution

It introduces a novel approach linking sampling in de Branges spaces to Naimark dilation, extending classical sampling theory to a broader functional framework.

Findings

Derived necessary and sufficient conditions for sampling sequences.

Connected sampling theory with Naimark dilation of frames.

Generalized classical Paley-Wiener sampling results.

Abstract

We consider the problem of sampling in de Branges spaces and develop some necessary conditions and some sufficient conditions for sampling sequences, which generalize some well-known sampling results in the Paley-Wiener space. These conditions are obtained by identifying the main construction with Naimark dilation of frames--embedding the de Branges space into a larger de Branges space while embedding the kernel functions associated with a sampling sequence into a Riesz basis for the larger space.