The smallest eigenvalue of the Hankel matrices associated with a perturbed Jacobi weight

Yuxi Wang and Yang Chen

Applied Mathematics and Computation, 2024, vol. 474, issue C

Abstract: In this paper, we study the large N behavior of the smallest eigenvalue λN of the (N+1)×(N+1) Hankel matrix, HN=(μj+k)0≤j,k≤N, generated by the γ dependent Jacobi weight w(z,γ)=e−γzzα(1−z)β,z∈[0,1],γ∈R,α>−1,β>−1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials PN(z), z∈C﹨[0,1], with the weight w(z,γ)=e−γzzα(1−z)β. Using the polynomials PN(z), we obtain the theoretical expression of λN, for large N. We also display the smallest eigenvalue λN for sufficiently large N, computed numerically.

Keywords: Orthogonal polynomials; Hankel matrices; Smallest eigenvalue; Asymptotics (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:474:y:2024:i:c:s0096300324000870

DOI: 10.1016/j.amc.2024.128615

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