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Two Variable Orthogonal Polynomials on the Bi-circle and Structured Matrices

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Jeffrey S. Geronimo, Hugo Woerdeman

Classical Analysis and ODEs · Submitted 14 Dec 2006

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

The authors' main claim These results are then used to construct a class of two variable measures supported on the bicircle that are given by one over the magnitude squared of a stable polynomial.

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arXiv IDmath/0612400 (version 1)
Pith IDG2XGQK62, Pith's permanent record of this paper and its review
CitationHugo Woerdeman, Jeffrey S. Geronimo. "Two Variable Orthogonal Polynomials on the Bi-circle and Structured Matrices." arXiv:math/0612400 (2006). https://pith.science/paper/G2XGQK62
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,464 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between the polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are derived and used to give necessary and sufficient conditions for the existence of a positive linear functional. These results are then used to construct a class of two variable measures supported on the bicircle that are given by one over the magnitude squared of a stable polynomial. Applications to Fej\'er-Riesz factorization are also given

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 We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. background 0.51 0.62 0.04 asserted 0.89
s1 The lexicographical and reverse lexicographical orderings are used to order the monomials. method 0.96 0.70 0.02 asserted 1.00
s2 Recurrence formulas are derived between the polynomials of different degrees. contribution 0.45 0.94 0.08 asserted 1.00
s3 These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. result 0.80 0.91 0.17 asserted 1.00
s4 Relations between the coefficients in the recurrence formulas are derived and used to give necessary and sufficient conditions for the existence of a positive linear functional. result 0.90 0.94 0.30 asserted 1.00
s5 These results are then used to construct a class of two variable measures supported on the bicircle that are given by one over the magnitude squared of a stable polynomial. result 0.55 0.92 0.28 asserted 1.00
s6 Applications to Fej\'er-Riesz factorization are also given contribution 0.78 0.87 0.07 asserted 0.99

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

We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between the polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are derived and used to give necessary and sufficient conditions for the existence of a positive linear functional. These results are then used to construct a class of two variable measures supported on the bicircle that are given by one over the magnitude squared of a stable polynomial. Applications to Fej\'er-Riesz factorization are also given

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