Sampling of operators

TL;DR

This paper extends classical sampling theory to operators by developing a sampling theorem for bandlimited operators, based on their Kohn-Nirenberg symbols, generalizing the concept from functions to operators.

Contribution

It introduces a novel sampling theorem for bandlimited operators, extending classical function sampling results to the operator setting using Kohn-Nirenberg symbols.

Findings

Sampling theorems for bandlimited operators are established.

Operators with bandlimited Kohn-Nirenberg symbols can be reconstructed from samples.

The results generalize classical sampling theorems for functions.

Abstract

Sampling and reconstruction of functions is a central tool in science. A key result is given by the sampling theorem for bandlimited functions attributed to Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling theory for operators which we call bandlimited if their Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for such operators and show that they are extensions of the classical sampling theorem.