Asymptotic Spectral Properties of the Hilbert \(L\) -Matrix
František Štampach · SIAM Journal on Matrix Analysis and Applications · 2022
Abstract. We study asymptotic spectral properties of the generalized Hilbert [Formula: see text]-matrix [Formula: see text] for large order [Formula: see text]. First, for general [Formula: see text], we deduce the asymptotic distribution of eigenvalues of [Formula: see text] outside the origin. Second, for [Formula: see text], asymptotic formulas for small eigenvalues of [Formula: see text] are derived. Third, in the classical case [Formula: see text], we also prove asymptotic formulas for large eigenvalues of [Formula: see text]. In particular, we obtain an asymptotic expansion of [Formula: see text] improving Wilf’s formula for the best constant in truncated Hardy’s inequality.