On determinant expansions for Hankel operators
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866914954890706944 |
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| author | Blower, Gordon Chen, Yang |
| author_facet | Blower, Gordon Chen, Yang |
| contents | Let $w$ be a semiclassical weight which is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. The paper expresses the Christoffel--Darboux kernel as a sum of products of Hankel integral operators. For $ψ\in L^\infty (i{\mathbb R})$, let $W(ψ)$ be the Wiener-Hopf operator with symbol $ψ$. The paper gives sufficient conditions on $ψ$ such that $1/\det W(ψ)W(ψ^{-1})=\det (I-Γ_{ϕ_1}Γ_{ϕ_2})$ where $Γ_{ϕ_1}$ and $Γ_{ϕ_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $ψ$, Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric ${}_nF_m$. These results extend those of Basor and Chen [2], who obtained ${}_4F_3$ likewise. The paper includes examples where the Wiener--Hopf factors are found explicitly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_05788 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On determinant expansions for Hankel operators Blower, Gordon Chen, Yang Functional Analysis Classical Analysis and ODEs 47B35 Let $w$ be a semiclassical weight which is generic in Magnus's sense, and $(p_n)_{n=0}^\infty$ the corresponding sequence of orthogonal polynomials. The paper expresses the Christoffel--Darboux kernel as a sum of products of Hankel integral operators. For $ψ\in L^\infty (i{\mathbb R})$, let $W(ψ)$ be the Wiener-Hopf operator with symbol $ψ$. The paper gives sufficient conditions on $ψ$ such that $1/\det W(ψ)W(ψ^{-1})=\det (I-Γ_{ϕ_1}Γ_{ϕ_2})$ where $Γ_{ϕ_1}$ and $Γ_{ϕ_2}$ are Hankel operators that are Hilbert--Schmidt. For certain $ψ$, Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric ${}_nF_m$. These results extend those of Basor and Chen [2], who obtained ${}_4F_3$ likewise. The paper includes examples where the Wiener--Hopf factors are found explicitly. |
| title | On determinant expansions for Hankel operators |
| topic | Functional Analysis Classical Analysis and ODEs 47B35 |
| url | https://arxiv.org/abs/1901.05788 |