Sharp results on sampling with derivatives in shift-invariant spaces and multi-window Gabor frames

TL;DR

This paper establishes precise conditions for sampling with derivatives in shift-invariant spaces generated by specific functions, and derives new results on multi-window Gabor frames, advancing understanding in sampling theory and frame analysis.

Contribution

It provides sharp Beurling density conditions for sampling with derivatives and introduces novel results on multi-window Gabor frames with special generating functions.

Findings

Sharp Beurling density conditions for sampling with derivatives

New results on multi-window Gabor frames with Hermite and totally positive functions

Enhanced understanding of sampling in shift-invariant spaces

Abstract

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.