A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants
In this paper, we will formulate 4×4 Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval [a, b], 0 < a < b < 1. The dist...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2020 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211020 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants. Roozbeh Gharakhloo and Alexander Its. SIGMA 16 (2020), 100, 47 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862647537856413696 |
|---|---|
| author | Gharakhloo, Roozbeh Its, Alexander |
| author_facet | Gharakhloo, Roozbeh Its, Alexander |
| citation_txt | A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants. Roozbeh Gharakhloo and Alexander Its. SIGMA 16 (2020), 100, 47 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we will formulate 4×4 Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval [a, b], 0 < a < b < 1. The distinguishing feature of this work is that in the formulation of the Riemann-Hilbert problem, no specific relationship is assumed between the Toeplitz and Hankel symbols. We will develop nonlinear steepest descent methods for analysing these problems in the case where the symbols are smooth (i.e., in the absence of Fisher-Hartwig singularities) and admit an analytic continuation in a neighborhood of the unit circle (if the symbol's support is the unit circle). We will finally introduce a model problem and will present its solution, requiring certain conditions on the ratio of Hankel and Toeplitz symbols. This, in turn, will allow us to find the asymptotics of the norms ₙ of the corresponding orthogonal polynomials and, in fact, the large asymptotics of the polynomials themselves. We will explain how this solvable case is related to the recent operator-theoretic approach in [Basor E., Ehrhardt T., in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Oper. Theory Adv. Appl., Vol. 259, Birkhäuser/Springer, Cham, 2017, 125-154, arXiv:1603.00506] to Toeplitz+Hankel determinants. At the end, we will discuss the prospects of future work and outline several technical, as well as conceptual, issues that we are going to address next within the 4×4 Riemann-Hilbert framework introduced in this paper.
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| first_indexed | 2026-03-15T12:32:10Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211020 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T12:32:10Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gharakhloo, Roozbeh Its, Alexander 2025-12-22T09:30:50Z 2020 A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants. Roozbeh Gharakhloo and Alexander Its. SIGMA 16 (2020), 100, 47 pages 1815-0659 2020 Mathematics Subject Classification: 15B05; 30E15; 35Q15 arXiv:1909.00963 https://nasplib.isofts.kiev.ua/handle/123456789/211020 https://doi.org/10.3842/SIGMA.2020.100 In this paper, we will formulate 4×4 Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval [a, b], 0 < a < b < 1. The distinguishing feature of this work is that in the formulation of the Riemann-Hilbert problem, no specific relationship is assumed between the Toeplitz and Hankel symbols. We will develop nonlinear steepest descent methods for analysing these problems in the case where the symbols are smooth (i.e., in the absence of Fisher-Hartwig singularities) and admit an analytic continuation in a neighborhood of the unit circle (if the symbol's support is the unit circle). We will finally introduce a model problem and will present its solution, requiring certain conditions on the ratio of Hankel and Toeplitz symbols. This, in turn, will allow us to find the asymptotics of the norms ₙ of the corresponding orthogonal polynomials and, in fact, the large asymptotics of the polynomials themselves. We will explain how this solvable case is related to the recent operator-theoretic approach in [Basor E., Ehrhardt T., in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Oper. Theory Adv. Appl., Vol. 259, Birkhäuser/Springer, Cham, 2017, 125-154, arXiv:1603.00506] to Toeplitz+Hankel determinants. At the end, we will discuss the prospects of future work and outline several technical, as well as conceptual, issues that we are going to address next within the 4×4 Riemann-Hilbert framework introduced in this paper. We are very grateful to Estelle Basor, Thomas Bothner, Christophe Charlier, Percy Deift, and Igor Krasovsky for their interest in this work and for many very useful comments and suggestions. We also thank the referees for their valuable remarks. R. Gharakhloo acknowledges support by NSF-grant DMS-1700261. A. It acknowledges support by NSF-grant DMS-1700261 and by Russian Science Foundation grant No. 17-11-01126. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants Article published earlier |
| spellingShingle | A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants Gharakhloo, Roozbeh Its, Alexander |
| title | A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants |
| title_full | A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants |
| title_fullStr | A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants |
| title_full_unstemmed | A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants |
| title_short | A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants |
| title_sort | riemann-hilbert approach to asymptotic analysis of toeplitz+hankel determinants |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211020 |
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