Dynamical sampling, derivatives, and interpolation formulas in the Paley--Wiener space

TL;DR

This paper develops criteria for interpolation formulas in Paley-Wiener space using Fourier multiplier operators, unifying and extending classical sampling and interpolation results for bandlimited functions.

Contribution

It introduces new criteria for interpolation in Paley-Wiener space based on Fourier multipliers, connecting and generalizing classical formulas.

Findings

Provides a unified framework for interpolation formulas

Recovers several classical sampling formulas as special cases

Extends interpolation techniques to broader classes of functions

Abstract

In this paper we present a criteria to obtain interpolations formulas in terms of the sequence $\left(\{T_n(f)(Nm)\}\}_{m\in\mathbb{Z}}\right)_{n=1}^N$, where $f$ are functions whose Fourier transform is supported in $[-1/2,1/2]$, and $T_n$ are certain Fourier multiplier operators. We also discuss applications and also prove that our results recover several classical formulas.