AI 中文总结
该研究针对局部紧阿贝尔群子群的平稳核，推导最小范数延拓算子的傅里叶表示，确定算子范数等性质，在紧群时延拓为压缩映射并对应高斯条件期望，还通过实例阐释理论。
AI 中文摘要
我们研究局部紧阿贝尔群$G$上平稳核到闭子群$H$的限制。对于非负谱密度$\tilde{k}$，我们推导了从$H$上受限核的再生核希尔伯特空间到$G$上原空间的最小范数延拓算子的显式纤维傅里叶表示；刻画了规范傅里叶公式何时能从$L^2(H)$有界延拓至$L^2(G)$，确定了其精确算子范数与下范数，并得到了相关插值空间的界。当$G$为紧群时，该延拓是压缩映射，对于具有可测连续版本的平稳高斯随机变量，可将观测到的限制映射为条件期望。我们还对先前断言的上确界范数压缩给出了反例，并通过环面一维子群上的基数插值与条件化阐释了该理论。
英文摘要
We study restrictions of stationary kernels on locally compact abelian groups $G$ to closed subgroups $H$. For a nonnegative spectral density $\hat k$, we derive an explicit fibrewise Fourier representation of the minimal-norm extension operator from the reproducing kernel Hilbert space of the restricted kernel on $H$ to the original space on $G$. We characterize when the canonical Fourier formula extends boundedly from $L^2(H)$ to $L^2(G)$, identify its exact operator norm and lower norm, and obtain bounds on the associated interpolation spaces. When $G$ is compact, the extension is a contraction and, for stationary Gaussian random variables admitting a measurable continuous version, maps the observed restriction to the conditional expectation. We also give a counterexample to a previously asserted supremum-norm contraction and illustrate the theory through cardinal interpolation and conditioning on one-dimensional subgroups of the torus.
