论文标题
希尔伯特$ l $ -matrix的渐近光谱特性
Asymptotic spectral properties of the Hilbert $L$-matrix
论文作者
论文摘要
我们研究广义Hilbert $ l $ -matrix \ [ l_ {n}（ν）= \ left（\ frac {1} {\ max（i，j）+ν} \ right）_ {i，j = 0}^{n-1}，\]，大订单$ n $。首先，对于一般$ν\ neq0，-1，-2，\ dots $，我们推断出$ l_ {n}（n}（v）$的渐近分布。其次，对于$ν> 0 $，得出了$ l_ {n}（ν）$的小特征值的渐近公式。第三，在经典案例$ν= 1 $中，我们还证明了$ l_ {n} \ equiv l_ {n}（1）$的大型特征值的渐近公式。我特别，我们获得了$ \ | l_ {n} \ | $改善Wilf公式的渐近扩展，以截断了Hardy的不平等中的最佳常数。
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 $ν\neq0,-1,-2,\dots$, we deduce the asymptotic distribution of eigenvalues of $L_{n}(ν)$ outside the origin. Second, for $ν>0$, asymptotic formulas for small eigenvalues of $L_{n}(ν)$ are derived. Third, in the classical case $ν=1$, we also prove asymptotic formulas for large eigenvalues of $L_{n}\equiv L_{n}(1)$. I particular, we obtain an asymptotic expansion of $\|L_{n}\|$ improving Wilf's formula for the best constant in truncated Hardy's inequality.
