Uniqueness and Reconstruction Theorems for Pseudodifferential Operators with a Bandlimited Kohn-Nirenberg Symbol

TL;DR

This paper develops a reconstruction formula and proves uniqueness theorems for pseudodifferential operators with bandlimited symbols, aiding channel estimation in wireless communications by linking operators to their Gabor matrix representations.

Contribution

It introduces a novel reconstruction formula and several uniqueness theorems connecting pseudodifferential operators to their Gabor matrix representations.

Findings

Reconstruction formula for bandlimited pseudodifferential operators.

Uniqueness theorems relating operators to Gabor matrices.

Enhanced understanding of operator identification in signal processing.

Abstract

Motivated by the problem of channel estimation in wireless communications, we derive a reconstruction formula for pseudodifferential operators with a bandlimited symbol. This reconstruction formula uses the diagonal entries of the matrix of the pseudodifferential operator with respect to a Gabor system. In addition, we prove several other uniqueness theorems that shed light on the relation between a pseudodifferential operators and its matrix with respect to a Gabor system.