Identification of minimum phase preserving operators on the half line

TL;DR

This paper solves the inverse problem of reconstructing minimum phase preserving operators on the half line using only two test functions, providing explicit formulas and revealing injectivity conditions.

Contribution

It provides a complete solution to reconstruct such operators from minimal data, leveraging recent advances in stable polynomial theory.

Findings

Reconstruction of operators from two test functions is possible.

Explicit integral representation of the operator is derived.

Operators with rank ≥ 2 are necessarily injective.

Abstract

Minimum phase functions are fundamental in a range of applications, including control theory, communication theory and signal processing. A basic mathematical challenge that arises in the context of geophysical imaging is to understand the structure of linear operators preserving the class of minimum phase functions. The heart of the matter is an inverse problem: to reconstruct an unknown minimum phase preserving operator from its value on a limited set of test functions. This entails, as a preliminary step, ascertaining sets of test functions that determine the operator, as well as the derivation of a corresponding reconstruction scheme. In the present paper we exploit a recent breakthrough in the theory of stable polynomials to solve the stated inverse problem completely. We prove that a minimum phase preserving operator on the half line can be reconstructed from data consisting of its value on precisely two test functions. And we derive an explicit integral representation of the unknown operator in terms of this data. A remarkable corollary of the solution is that if a linear minimum phase preserving operator has rank at least two, then it is necessarily injective.