We study asymptotic spectral properties of the generalized Hilbert L-matrix
Ln(ν)=(max(i,j)+ν1)i,j=0n−1, for large
order n. First, for general ν=0,−1,−2,…, we deduce the asymptotic
distribution of eigenvalues of Ln(ν) outside the origin. Second, for
ν>0, asymptotic formulas for small eigenvalues of Ln(ν) are derived.
Third, in the classical case ν=1, we also prove asymptotic formulas for
large eigenvalues of Ln≡Ln(1). I particular, we obtain an
asymptotic expansion of ∥Ln∥ improving Wilf's formula for the best
constant in truncated Hardy's inequality.Comment: 18 pages, dedicated to the memory of Harold Widom, accepted for
publication in SIAM J. Matrix Anal. App