论文标题
Szegő限制了与希尔伯特矩阵有关的定理
A Szegő Limit Theorem Related to the Hilbert Matrix
论文作者
论文摘要
事实证明，在特殊情况下，$ 1- \fracβπh_{n，α} $，其中$ h_ {n，α} $是$ n \ times n \ times n \ times n $ -hilbert matrix，$ n $ n $α\ geq frac {2 $β\ in \ mathbf {c} $。该证明使用操作者理论工具，并减少了Carleman操作员的经典KAC-Akhiezer定理。因此，该特殊Hankel矩阵的定理的有效性可以从$ |β| <1 $扩展到\ Mathbf {c} \ setMinus] 1，\ infty [$。校正期限的限制将提高到$ o（1）$，而不是$ o（\ ln（n））$ for $β\ in \ mathbf {c} \ setMinus [1，\ infty [$。极限情况$β= 1 $直接从$β$的一般渐近学中得出。
The Szegő limit theorem by Fedele and Gebert for matrices of the type identity minus Hankel matrix is proved for the special case $1-\fracβπH_{N,α}$ where $H_{N,α}$ is the $N\times N$-Hilbert matrix, $α\geq\frac{1}{2}$, and $β\in\mathbf{C}$. The proof uses operator theoretic tools and a reduction to the classical Kac--Akhiezer theorem for the Carleman operator. Thereby, the validity of the theorem for this special Hankel matrix can be extended from $|β|<1$ to $β\in\mathbf{C}\setminus ]1,\infty[$. The bound on the correction term is improved to $O(1)$ instead of $o(\ln(N))$ for $β\in\mathbf{C}\setminus [1,\infty[$. The limit case $β=1$ is derived directly from the asymptotics for general $β$.
