arXiv subjects

Charles Weems

Publications and source records attributed to Charles Weems.

2 recordsLinked to original sources

arXiv2018

The Smallest Eigenvalue of Large Hankel Matrices

We investigate the large $N$ behavior of the smallest eigenvalue, $λ_{N}$, of an $\left(N+1\right)\times \left(N+1\right)$ Hankel (or moments) matrix $\mathcal{H}_{N}$, generated by the weight $w(x)=x^α(1-x)^β,~x\in[0,1],~ α>-1,~β>-1$. By applying the arguments of Szegö, 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 $λ_{N}$. Based on this formula, we produce the expressions for $λ_{N}$, 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 $N$.

math-ph↗

arXiv2018

The Smallest Eigenvalue of Large Hankel Matrices Generated by a Deformed Laguerre Weight

We study the asymptotic behavior of the smallest eigenvalue, $λ_{N}$, of the Hankel (or moments) matrix denoted by $\mathcal{H}_{N}=\left(μ_{m+n}\right)_{0\leq m,n\leq N}$, with respect to the weight $w(x)=x^α{\rm e}^{-x^β},~x\in[0,\infty),~α>-1,~β>\frac{1}{2}$. Based on the research by Szegö, Chen, etc., we obtain an asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with $w(x)$. Using this, we obtain the specific asymptotic formulas of $λ_{N}$ in this paper. Applying the parallel algorithm discovered by Emmart, Chen and Weems, we get a variety of numerical results of $λ_{N}$ corresponding to our theoretical calculations.

math-ph↗