arXiv 1807.014412018-07-04math.FA

具有某些Piecewise Continuous符号的Toeplitz矩阵的谱渐近性

Spectral Asymptotics for Toeplitz Matrices Having Certain Piecewise Continuous Symbols

Richard A. Libby

备注：: 20 pages, 0 figures, presented at the Harold Widom Anniversary Workshop on Toeplitz and Wiener-Hopf Operators, Santa Cruz, California, September 1992

摘要：

研究了单位圆$\mathbb{T}$上定义的复值函数$σ$的Toeplitz矩阵$T_{n}[σ]=(\hatσ(i-j))$特征值的极限行为，其中$0\leq i,j \leq n$为$n \to \infty$.当$σ(z)=(-z)^βτ(z)$，$β$不为整数且$τ$满足一定的光滑条件时，则$\det T_{n}[σ]=\mathbf{G}[τ]^{n+1}n^{-β^{2}}E[τ,β](1+o(1))$为$n \to \infty$，其中$\mathbf{G}[τ]$表示$τ$的几何平均，$E$为与$n$无关的常数. $E$的值被发现在$τ$的傅立叶系数和$β$的解析函数。这些结果以前在$\Re β$（$β$的实部）足够小的情况下是已知的。这个结果的一个推论是确定的极限集和极限分布的特征值的$T_{n}[σ]$。

英文摘要：

The limiting behavior of the eigenvalues of the Toeplitz matrices $T_{n}[σ]=(\hatσ(i-j))$, where $0\leq i,j \leq n$, as $n \to \infty$, is investigated in the case of complex valued functions $σ$ defined on the unit circle $\mathbb{T}$ and having exactly one point of discontinuity. It is found that if $σ(z)=(-z)^βτ(z)$, $β$ not an integer and $τ$ satisfying certain smoothness conditions, then $\det T_{n}[σ]=\mathbf{G}[τ]^{n+1}n^{-β^{2}}E[τ,β](1+o(1))$ as $n \to \infty$, where $\mathbf{G}[τ]$ denotes the geometric mean of $τ$ and $E$ is a constant independent of $n$. A value for $E$ is found in terms of the Fourier coefficients of $τ$ and an analytic function of $β$. These results were known previously in the case that $\Re β$, the real part of $β$, was sufficiently small. A corollary of this result is a determination of the limiting set and limiting distributions for the eigenvalues of $T_{n}[σ]$.

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英文摘要：

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