正定Toeplitz矩阵范数的渐近估计与平稳随机信号拟周期分量的检测
Asymptotic estimates of the norms of positive definite Toeplitz matrices and detection of quasi-periodic components of stationary random signals
摘要：
确定了正定Toeplitz矩阵$Q_{N}=(b(j-k))_{j,k=0}^{N-1}$为$N\to \infty $的Hilbert-Scandal范数和Hilbert范数的渐近形式.这里$b(j)$是在$[ -π,π] $上生成非负测量$dσ(θ)$的后向三角矩。证明了$σ(θ)$是连续的当且仅当这些贡献中的任何一个是$o(N)$。给出了具有差分核的正积分算子的类似判别准则。所得结果适用于处理平稳随机信号，特别是沸水核反应堆发出的中子信号。
英文摘要：
Asymptotic forms of the Hilbert-Scmidt and Hilbert norms of positive definite Toeplitz matrices $Q_{N}=(b(j-k))_{j,k=0}^{N-1}$ as $N\to \infty $ are determined. Here $b(j)$ are consequent trigonometric moments of a generating non-negative mesure $dσ(θ)$ on $[ -π,π] $. It is proven that $σ(θ)$ is continuous if and only if any of those contributions is $o(N)$. Analogous criteria are given for positive integral operators with difference kernels. Obtained results are applied to processing of stationary random signals, in particular, neutron signals emitted by boiling water nuclear reactors.
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