Hankel算子奇异值的精确估计
Sharp estimates for singular values of Hankel operators
摘要：
我们考虑在$\ell^2(\mathbb Z_+)$中实现的紧Hankel算子为具有矩阵元$h(j+k)$的无限矩阵$Γ$。粗略地说，我们证明了，对于所有的$α>0$，$Γ$的奇异值$s_{n}$满足约束$s_{n}= O(n^{-α})$作为$n\to \infty$提供$h(j)= O(j^{-1}(\log j)^{-α})$作为$j\to \infty$。这些对$s_{n}$的估计在$α$的功率标度中是尖锐的。对于在$L^2(\mathbb R_+)$中实现的Hankel算子$\mathbfΓ$作为具有核$\mathbf h(t+s)$的积分算子得到了类似的结果。在这种情况下，$\mathbfΓ$的奇异值的估计由$\mathbf h(t)$作为$t\to 0$和作为$t\to\infty$的行为确定。
英文摘要：
We consider compact Hankel operators realized in $\ell^2(\mathbb Z_+)$ as infinite matrices $Γ$ with matrix elements $h(j+k)$. Roughly speaking, we show that, for all $α>0$, the singular values $s_{n}$ of $Γ$ satisfy the bound $s_{n}= O(n^{-α})$ as $n\to \infty$ provided $h(j)= O(j^{-1}(\log j)^{-α})$ as $j\to \infty$. These estimates on $s_{n}$ are sharp in the power scale of $α$. Similar results are obtained for Hankel operators $\mathbfΓ$ realized in $L^2(\mathbb R_+)$ as integral operators with kernels $\mathbf h(t+s)$. In this case the estimates of singular values of $\mathbfΓ$ are determined by the behavior of $\mathbf h(t)$ as $t\to 0$ and as $t\to\infty$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
