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Peter C. Gibson

Publications and source records attributed to Peter C. Gibson.

16 recordsLinked to original sources
arXiv2025

Integration of an arbitrary linear ODE

The standard text book theory of ODEs lacks a general method to solve linear equations having variable coefficients, providing instead a collection of special techniques for particular classes of equations. The present article addresses this shortcoming in the basic theory. We introduce the multex integral operator, generalizing to several input functions the standard exponential primitive operator that is inverse to the logarithmic derivative. The multex operator serves to integrate in explicit form an arbitrary linear ordinary differential equation.

math.CA↗
arXiv2025

Solution of the scalar Riccati equation

The scalar Riccati equation is a prototypical nonlinear ODE having diverse mathematical connections. In the centuries since its initial formulation, a standard textbook theory has emerged according to which the general solution may be determined if a particular solution is known; but no general method exists to determine a particular solution explicitly, except in sporadic special cases. The purpose of the present article is to solve the scalar Riccati equation in general form, as well as the general linear ODE of second order, directly by explicit construction. In the case of the Riccati equation, the solution sets up a bijective correspondence between triples of locally integrable functions on the real line, and locally absolutely continuous paths through the identity in the automorphism group of the Riemann sphere. As applications of the results, we obtain an explicit solution to the one-dimensional Schrödinger equation, an inversion formula for the Miura transform, and a new formula for Airy functions.

math.CA↗
arXiv2022

Scattering on the line via singular approximation

Motivated by applications to acoustic imaging, the present work establishes a framework to analyze scattering for the one-dimensional wave, Helmholtz, Schrödinger and Riccati equations that allows for coefficients which are more singular than can be accommodated by previous theory. In place of the standard scattering matrix or the Weyl-Titchmarsh $m$-function, the analysis centres on a new object, the generalized reflection coefficient, which maps frequency (or the spectral parameter) to automorphisms of the Poincaré disk. Purely singular versions of the generalized reflection coefficient, which are amenable to direct analysis, serve to approximate the general case. Orthogonal polynomials on both the unit circle and unit disk play a key technical role, as does an exotic Riemannian structure on PSL$(2,\mathbb{R})$. A central role is also played by the newly-defined harmonic exponential operator, introduced to mediate between impedance (or index of refraction) and the reflection coefficient. The approach leads to new, explicit formulas and effective algorithms for both forward and inverse scattering. The algorithms may be viewed as nonlinear analogues of the FFT. In addition, the scattering relation is shown to be elementary in a precise sense at or below the critical threshold of continuous impedance. For discontinuous impedance, however, the reflection coefficient ceases to decay at infinity, the classical trace formula breaks down, and the scattering relation is complicated by the emergence of almost periodic structure.

math.AP↗
arXiv2022

Theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings

The classical Szegő-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal polynomials determined by the measure. The present paper constructs orthogonal polynomials on the torus of arbitrary finite dimension in order to prove theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings. The results are applied to the one-dimensional Schrödinger equation in impedance form to yield a new trace formula valid for piecewise constant impedance, a case where the classical trace formula breaks down. As a byproduct, the analysis gives an explicit formula for the Taylor coefficients of a bounded holomorphic function on the open disk in terms of its continued fraction expansion.

math.FA↗
arXiv2019

Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies and the jump points of the wave speed are equally spaced with respect to travel time. Numerical examples show that the algorithm works also in the general case of arbitrary wave speed (either with jumps or continuously varying etc.) giving progressively more accurate approximations as the range of recorded frequencies increases. A key underlying theoretical insight is to associate scattering data to compositions of automorphisms of the unit disk, which are in turn related to orthogonal polynomials on the unit circle. The algorithm exploits the three-term recurrence of orthogonal polynomials to reduce the required computation.

math.AP↗
arXiv2018

The refined impedance transform for 1D acoustic reflection data

The one dimensional wave equation serves as a basic model for imaging modalities such as seismic which utilize acoustic data reflected back from a layered medium. In 1955 Peterson et al. described a single scattering approximation for the one dimensional wave equation that relates the reflection Green's function to acoustic impedance. The approximation is simple, fast to compute and has become a standard part of seismic theory. The present paper re-examines this classical approximation in light of new results concerning the (exact) measurement operator for reflection imaging of layered media, and shows that the classical approximation can be substantially improved. We derive an alternate formula, called the refined impedance transform, that retains the simplicity and speed of computation of the classical estimate, but which is qualitatively more accurate and applicable to a wider range of recorded data. The refined impedance transform can be applied to recorded data directly (without the need to deconvolve the source wavelet), and solves exactly the inverse problem of determining the value of acoustic impedance on the far side of an arbitrary slab of unknown structure. The results are illustrated with numerical examples.

math.AP↗
arXiv2016

Fourier expansion of disk automorphisms via scattering in layered media

A family of orthogonal polynomials on the disk (which we call scattering polynomials) serves to formulate a remarkable Fourier expansion of the composition of a sequence of Poincaré disk automorphisms. Scattering polynomials are tied to an exotic riemannian structure on the disk that is hybrid between hyperbolic and euclidean geometries, and the expansion therefore links this exotic structure to the usual hyperbolic one. The resulting identity is intimately connected with the scattering of plane waves in piecewise constant layered media. Indeed, a recently established combinatorial analysis of scattering sequences provides a key ingredient of the proof. At the same time, the polynomial obtained by truncation of the Fourier expansion elegantly encodes the structure of the nonlinear measurement operator associated with the finite time duration scattering experiment.

math.AP↗
arXiv2015

Constructive solutions to Pólya-Schur problems

We present constructive solutions to the following Pólya-Schur problems concerning linear operators on the space of univariate polynomials: Given subsets $Ω_1$ and $Ω_2$ of the complex plane, determine operators that map all polynomials having no zeros in $Ω_1$ to polynomials having no zeros in $Ω_2$, or to the zero polynomial. We describe an explicit class consisting of rank 1 operators and product-composition operators that solve the stated problems for arbitrary $Ω_1$ and $Ω_2$; and this class is shown to comprise all solutions when $Ω_1$ is bounded and $Ω_2$ has non-empty interior. The latter result encompasses a number of open problems and, moreover, gives explicit solutions in cases of circular domains $Ω_1=Ω_2$ where existing characterizations are non-constructive. The paper also treats problems stemming from digital signal processing that are analogous to Pólya-Schur problems. Specifically, we describe all bounded linear operators on Hardy space that preserve the class of outer functions, as well as those that preserve shifted outer functions.

math.CV↗
arXiv2014

A smooth global model for scattering in layered media

Layered media have been studied extensively both for their importance in imaging technologies and as an example of a hyperbolic PDE with discontinuous coefficients. From the perspective of acoustic imaging, the time limited impulse response at the boundary, or boundary Green's function, represents measured data, and the objective is to determine coefficients, which encode physical parameters, from the data. The present paper resolves two fundamental open problems for layered media: (1) how to compute the time limited Green's function in the presence of discontinuous coefficients; and (2) to determine precisely how data depends on coefficients. We show that there exists a single system of equations in $3n$-dimensional space that governs the parameterized family of all $n$-layered media simultaneously. The alternate system has smooth coefficients, can be solved directly by separation of variables, and recovers the impulse response at the boundary for the original equations. The analysis brings to light an exotic laplacian---hybrid between the euclidean and hyperbolic laplacians---that plays a central role in the scattering process. Its eigenfunctions comprise a new family of orthogonal polynomials on the disk. These serve as building blocks for a universal wavefield in terms of which the dependence of data on coefficients has a simple description: reflection data is obtained by sampling a translate of the wavefield on the integer lattice and then pushing forward by a linear functional, where the translate and pushforward correspond to reflectivity and layer depth vectors, respectively.

math.AP↗
arXiv2013

Hardy space on the polydisk and scattering in layered media

Hardy space on the polydisk provides the setting for a global description of scattering in piecewise-constant layered media, giving a simple qualitative interpretation for the nonlinear dependence of the Green's function on reflection coefficients and layer depths. Using explicit formulas for amplitudes, we prove that the power spectrum of the Green's function is approximately constant. In addition we exploit a connection to Jacobi polynomials to derive formulas for computing reflection coefficients from partial amplitude data. Unlike most approaches to layered media, which variously involve scaling limits, approximations or iterative methods, the formulas and methods in the present paper are exact and direct.

math-ph↗
arXiv2013

The combinatorics of scattering in layered media

Reflection and transmission of waves in piecewise constant layered media are important in various imaging modalities and have been studied extensively. Despite this, no exact time domain formulas for the Green's functions have been established. Indeed, there is an underlying combinatorial obstacle: the analysis of scattering sequences. In the present paper we exploit a representation of scattering sequences in terms of trees to solve completely the inherent combinatorial problem, and thereby derive new, explicit formulas for the reflection and transmission Green's functions.

math.CO↗
arXiv2012

The purely singular 1-D acoustic reflection problem

This paper analyzes the nonlinear correspondence between the reflectivity profile (model) and the plane wave impulse response at the boundary (data) for a three-dimensional half space consisting of a sequence of homogeneous horizontal layers. This correspondence is of importance in geophysical imaging, where it has been studied for more than half a century from a variety of perspectives. The main contribution of the present paper is to derive something new in the context of a time-limited deterministic approach: (i) an exact finite (non-asymptotic) formula for the data in terms of the model, (ii) a corresponding exact inverse algorithm, and (iii) a precise characterization of the inherent nonlinearity. Regarding (iii), for generic models the correspondence is characterized as a pair of maps, one of which is locally linear, and the other of which is locally polynomial. Both are determined by a local combinatorial invariant, an integer matrix. Concerning (ii), the basic inverse algorithm is modified to allow for erroneous amplitude data, taking advantage of the overdeterminacy of the inverse problem to recover the exact model even in cases where the data is badly distorted. The results are illustrated with numerical examples.

math-ph↗
arXiv2012

The structure of test functions that determine weighted composition operators

In the context of analytic functions on the open unit disk, a weighted composition operator is simply a composition operator followed by a multiplication operator. The class of weighted composition operators has an important place in the theory of Banach spaces of analytic functions; for instance, it includes all isometries on $H^p$ $(p\neq 2)$. Very recently it was shown that only weighted composition operators preserve the class of outer functions. The present paper considers a particular question motivated by applications: Which smallest possible sets of test functions can be used to identify an unknown weighted composition operator? This stems from a practical problem in signal processing, where one seeks to identify an unknown minimum phase preserving operator on $L^2(\real_+)$ using test signals. It is shown in the present paper that functions that determine weighted composition operators are directly linked to the classical normal family of schlicht functions. The main result is that a pair of functions $\{f,g\}$ distinguishes between any two weighted composition operators if and only if there exists a zero-free function $h$ and a schlicht function $σ$ such that $\spn\{f,g\}=\spn\{hσ,h\}$. This solves completely the underlying signal processing problem and brings to light an intriguing geometric object, the manifold of planes of the form $\spn\{hσ,h\}$. As an application of the main result, it is proven that there exist compactly supported pairs in $L^2(\real_+)$ that can be used to identify minimum phase preserving operators.

math.FA↗
arXiv2011

Identification of minimum phase preserving operators on the half line

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.

math.FA↗