Home > Research > Publications & Outputs > On determinant expansions for Hankel operators

Research

Associated organisational unit

Electronic data

  • Barnes15January20

    Accepted author manuscript, 429 KB, PDF document

    Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License

Links

Text available via DOI:

Keywords

View graph of relations

On determinant expansions for Hankel operators

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

Standard

On determinant expansions for Hankel operators. / Blower, Gordon; Chen, Yang.
In: Concrete Operators, Vol. 7, No. 1, 04.02.2020, p. 13-44.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Blower, G & Chen, Y 2020, 'On determinant expansions for Hankel operators', Concrete Operators, vol. 7, no. 1, pp. 13-44. https://doi.org/10.1515/conop-2020-0002

APA

Blower, G., & Chen, Y. (2020). On determinant expansions for Hankel operators. Concrete Operators, 7(1), 13-44. https://doi.org/10.1515/conop-2020-0002

Vancouver

Blower G, Chen Y. On determinant expansions for Hankel operators. Concrete Operators. 2020 Feb 4;7(1):13-44. doi: 10.1515/conop-2020-0002

Author

Blower, Gordon ; Chen, Yang. / On determinant expansions for Hankel operators. In: Concrete Operators. 2020 ; Vol. 7, No. 1. pp. 13-44.

Bibtex

@article{56f443140c5f4b5d92bbc450636b6fcf,
title = "On determinant expansions for Hankel operators",
abstract = "Let $w$ be a semiclassical weight that is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. We express the Christoffel--Darboux kernel as a sum of products of Hankel integral operators. For $\psi\in L^\infty (i{\mathbb R})$, let $W(\psi )$ be the Wiener-Hopf operator with symbol $\psi$. We give sufficient conditions on $\psi$ such that $1/\det W(\psi )W(\psi^{-1})=\det (I-\Gamma_{\phi_1}\Gamma_{\phi_2})$ where $\Gamma_{\phi_1}$ and $\Gamma_{\phi_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $\psi$, Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric ${}_{2m}F_{2m-1}$. These results extend those of Basor and Chen \cite{BasorChen2003}, who obtained ${}_4F_3$ likewise. We include examples where the Wiener--Hopf factors are found explicitly. \par\vskip.1in",
keywords = "orthogonal polynomials, Wiener-Hopf factorization, special functions, linear systems",
author = "Gordon Blower and Yang Chen",
year = "2020",
month = feb,
day = "4",
doi = "10.1515/conop-2020-0002",
language = "English",
volume = "7",
pages = "13--44",
journal = "Concrete Operators",
issn = "2299-3282",
publisher = "de Gruyter",
number = "1",

}

RIS

TY - JOUR

T1 - On determinant expansions for Hankel operators

AU - Blower, Gordon

AU - Chen, Yang

PY - 2020/2/4

Y1 - 2020/2/4

N2 - Let $w$ be a semiclassical weight that is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. We express the Christoffel--Darboux kernel as a sum of products of Hankel integral operators. For $\psi\in L^\infty (i{\mathbb R})$, let $W(\psi )$ be the Wiener-Hopf operator with symbol $\psi$. We give sufficient conditions on $\psi$ such that $1/\det W(\psi )W(\psi^{-1})=\det (I-\Gamma_{\phi_1}\Gamma_{\phi_2})$ where $\Gamma_{\phi_1}$ and $\Gamma_{\phi_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $\psi$, Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric ${}_{2m}F_{2m-1}$. These results extend those of Basor and Chen \cite{BasorChen2003}, who obtained ${}_4F_3$ likewise. We include examples where the Wiener--Hopf factors are found explicitly. \par\vskip.1in

AB - Let $w$ be a semiclassical weight that is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. We express the Christoffel--Darboux kernel as a sum of products of Hankel integral operators. For $\psi\in L^\infty (i{\mathbb R})$, let $W(\psi )$ be the Wiener-Hopf operator with symbol $\psi$. We give sufficient conditions on $\psi$ such that $1/\det W(\psi )W(\psi^{-1})=\det (I-\Gamma_{\phi_1}\Gamma_{\phi_2})$ where $\Gamma_{\phi_1}$ and $\Gamma_{\phi_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $\psi$, Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric ${}_{2m}F_{2m-1}$. These results extend those of Basor and Chen \cite{BasorChen2003}, who obtained ${}_4F_3$ likewise. We include examples where the Wiener--Hopf factors are found explicitly. \par\vskip.1in

KW - orthogonal polynomials

KW - Wiener-Hopf factorization

KW - special functions

KW - linear systems

U2 - 10.1515/conop-2020-0002

DO - 10.1515/conop-2020-0002

M3 - Journal article

VL - 7

SP - 13

EP - 44

JO - Concrete Operators

JF - Concrete Operators

SN - 2299-3282

IS - 1

ER -