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arXiv · 1803.11324

The Smallest Eigenvalue of Large Hankel Matrices

Mengkun Zhu·Yang Chen·Niall Emmart·Charles Weems
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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 $N$.

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Mengkun Zhu, Yang Chen, Niall Emmart, Charles Weems. 2018-03-30. The Smallest Eigenvalue of Large Hankel Matrices. https://arxiv.org/abs/1803.11324

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