The smallest eigenvalue of large Hankel matrices

摘要

We investigate the large N behavior of the smallest eigenvalue, λN, of an (N+1)×(N+1) Hankel (or moments) matrix HN, generated by the weight w(x)=xα(1−x)β,x∈[0,1],α>−1,β>−1. By applying the arguments of Szegö, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials Pn(z),z∈C∖[0,1], associated with w(x), which are required in the determination of λN. Based on this formula, we produce the expressions for λN, 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.