Sharp results on sampling with derivatives in bandlimited functions

TL;DR

This paper establishes sharp maximum gap conditions for sampling and reconstructing bandlimited functions using derivatives, ensuring uniqueness and stability in nonuniform sampling scenarios.

Contribution

It provides the first sharp maximum gap conditions for sampling with derivatives in bandlimited functions, extending classical sampling theory.

Findings

Reconstruction is unique and stable if maximum gap < c.

Sets with maximum gap ≤ c are uniqueness sets for derivatives.

Sharp maximum gap condition for first derivatives is obtained.

Abstract

We discuss the problems of uniqueness, sampling and reconstruction with derivatives in the space of bandlimited functions. We prove that if X is sequence of real numbers such that the maximum gap between two consecutive samples is less than certain positive constant c, then bandlimited function of bandwidth {\sigma} can be reconstructed uniquely and stably from its nonuniform samples involving derivatives. We also prove that if the maximum gap is less than or equal to c, then X is a set of uniqueness for the space of bandlimited functions of bandwidth {\sigma} when the samples involving k-1 derivatives. As a by-product we obtain the sharp maximum gap condition for samples involving first derivatives.