Asymptotic spectral properties of the Hilbert $L$-matrix

Abstract

We study asymptotic spectral properties of the generalized Hilbert $L$-matrix \[ L_{n}(ν)=\left(\frac{1}{\max(i,j)+ν}\right)_{i,j=0}^{n-1}, \] for large order $n$. First, for general $\nu \ne 0,-1,-2,\dots$, we deduce the asymptotic distribution of eigenvalues of $L_n(\nu )$ outside the origin. Second, for $\nu >0$, asymptotic formulas for small eigenvalues of $L_n(\nu )$ are derived. Third, in the classical case $\nu =1$, we also prove asymptotic formulas for large eigenvalues of $L_n\equiv L_n(1)$. I particular, we obtain an asymptotic expansion of $\Vert L_n\Vert$ improving Wilf's formula for the best constant in truncated Hardy's inequality.

arXiv comment18 pages, dedicated to the memory of Harold Widom, accepted for publication in SIAM J. Matrix Anal. Appl