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Fourier expansion of disk automorphisms via scattering in layered media

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Peter C. Gibson

Analysis of PDEs · Submitted 7 Mar 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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 Poincar\'e disk automorphisms.

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arXiv ID1603.02304 (version 1)
Pith IDNGRJBQ6I, Pith's permanent record of this paper and its review
CitationPeter C. Gibson. "Fourier expansion of disk automorphisms via scattering in layered media." arXiv:1603.02304 (2016). https://pith.science/paper/NGRJBQ6I
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,322 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.05 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.20 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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 Poincar\'e 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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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 Poincar\'e disk automorphisms. contribution 0.83 0.86 0.69 asserted 1.00
s1 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. interpretation 0.91 0.63 0.19 asserted 1.00
s2 The resulting identity is intimately connected with the scattering of plane waves in piecewise constant layered media. interpretation 0.70 0.55 0.22 asserted 1.00
s3 Indeed, a recently established combinatorial analysis of scattering sequences provides a key ingredient of the proof. method 0.77 0.10 0.04 asserted 1.00
s4 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. result 0.56 0.78 0.19 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

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 Poincar\'e 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.

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