Outer preserving linear operators

Gibson, Peter C.; Lamoureux, Michael P.; Margrave, Gary F.

doi:10.1016/j.jfa.2011.07.003

Cited by 5 publications

(13 citation statements)

References 6 publications

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“…The mapping d does not carry translates of continuous signals to translates of discrete signals, so by contrast Theorem 6 cannot be proved using its discrete analogue. Indeed, whereas Theorem 6 needs the full force of Theorem 2, its discrete analogue was first proved in [1,Theorem 4] using less powerful methods.…”

Section: Operators Preserving Mmentioning

confidence: 99%

Identification of minimum-phase-preserving operators on the half-line

Gibson¹,

Lamoureux²

2012

Inverse Problems

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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.

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Section: Operators Preserving Mmentioning

confidence: 99%

Identification of minimum-phase-preserving operators on the half-line

Gibson¹,

Lamoureux²

2012

Inverse Problems

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“…Independently of the above developments, we have studied an analytic version of a Pólya-Schur problem in the context of the Hardy-Hilbert space H 2 = H 2 (D) on the unit disk, motivated by geophysical applications [5]. More precisely, the classical factorization theorem for H 2 expresses an arbitrary function as a product of an inner function, defined to have constant modulus 1 on the boundary circle, and an outer function, by definition a cyclic vector of the unilateral shift (see [9]).…”

Section: Introductionmentioning

confidence: 99%

“…Thus a function f ∈ H 2 is outer if and only if the span of functions of the form z n f (z), where n ≥ 0, is dense in H 2 . Referring to functions of the form z n f (z) as shifted outer functions, [5] constructively characterizes continuous linear operators A : H 2 → H 2 that preserve the class O ⊂ H 2 of all shifted outer functions. This is analogous to a Pólya-Schur problem because outer functions have no zeros in the open unit disk D, and hence shifted outer functions have no zeros in the punctured disk D \ {0}.…”

Section: Introductionmentioning

confidence: 99%

“…This is analogous to a Pólya-Schur problem because outer functions have no zeros in the open unit disk D, and hence shifted outer functions have no zeros in the punctured disk D \ {0}. The characterization of operators that preserve shifted outer functions translates via the inverse Z-transform to a characterization of operators acting on causal digital signals that preserve the class of delayed minimum-phase signals, a property of importance for seismic recordings (see [5] for full details).…”

Section: Introductionmentioning

confidence: 99%

“…The present paper stems from two principal objectives: (i) to bring the classical Pólya-Schur problems and their analytic analogues together within a common framework; and (ii) to extend the characterization of operators preserving delayed minimum-phase digital signals to operators acting on continuous signals in L 2 (R + ), where the methods of [5] are insufficient. We prove a structure theorem that achieves both of these objectives and that furthermore solves an array of previously open Pólya-Schur problems.…”

Section: Introductionmentioning

confidence: 99%

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Constructive solutions to Pólya–Schur problems

Gibson

Lamoureux

2015

Journal of Functional Analysis

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The general Pólya-Schur problem is to characterize linear operators on the space of univariate polynomials that preserve stability, where a polynomial is stable with respect to a region Ω in the complex plane if it has no zeros in Ω. Stable preserving operators have proven to be important in a variety of applications ranging from statistical mechanics to combinatorics, and variants of Pólya-Schur problems involving analytic functions are important in applications to signal processing. We present a structure theorem that bridges polynomial and analytic Pólya-Schur problems, providing constructive characterizations of stable-preserving operators for a general class of domains Ω. The structure theorem facilitates the solution of open Pólya-Schur problems in the classical setting, and provides constructive characterizations of stable preserving operators in cases where previously known characterizations are non-constructive. In the analytic setting, the structure theorem enables the explicit characterization of minimum-phase preserving operators on the half-line, a problem of importance in geophysical signal processing.

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Which linear operators preserve outer functions?

Kou

Liu

2017

Indagationes Mathematicae

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Outer preserving linear operators

Cited by 5 publications

References 6 publications

Identification of minimum-phase-preserving operators on the half-line

Identification of minimum-phase-preserving operators on the half-line

Constructive solutions to Pólya–Schur problems

Which linear operators preserve outer functions?

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