The Smallest Eigenvalue of Large Hankel Matrices

TL;DR

This paper derives the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by a specific weight function, combining theoretical analysis with numerical validation.

Contribution

It provides a new asymptotic formula for the smallest eigenvalue of large Hankel matrices with a specific weight, extending previous theoretical methods.

Findings

Asymptotic formula for $oldsymbol{ ext{λ}_N}$ derived

Theoretical results closely match numerical simulations for large N

Method applicable to Hankel matrices with similar weight functions

Abstract

We investigate the large $N$ behavior of the smallest eigenvalue, $\lambda_{N}$, of an $\left(N+1\right)\times \left(N+1\right)$ Hankel (or moments) matrix $\mathcal{H}_{N}$, generated by the weight $w(x)=x^{\alpha}(1-x)^{\beta},~x\in[0,1],~ \alpha>-1,~\beta>-1$. By applying the arguments of Szeg\"{o}, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials $P_{n}(z),z\in\mathbb{C}\setminus[0,1]$, associated with $w(x)$, which are required in the determination of $\lambda_{N}$. Based on this formula, we produce the expressions for $\lambda_{N}$, for large $N$. Using the parallel algorithm presented by Emmart, Chen and Weems, we show that the theoretical results are in close proximity to the numerical results for sufficiently large $N$.