局部紧群上Orlicz与混合范数Orlicz空间中非衰减信号的相关采样

中文摘要 AI 辅助

我们研究了定义在局部紧群上的非衰减信号的采样问题。信号被建模为加权Orlicz空间和加权混合范数Orlicz空间的适当子空间的元素，从而允许在无穷远处有受控的增长。我们的主要焦点是幂等积分算子的像空间。在相关积分核满足适当的振荡估计条件下，我们建立了确定性采样定理，保证从点采样中稳定重建。我们还在相同背景下证明了一个平均采样结果。此外，我们证明了范数本质上集中在群的紧子集上的函数的随机采样定理。特别地，我们表明，以高概率，可以从$\mathcal{O}\left(\mu(K)\log \mu(K)\right)$个随机选择的采样点进行采样，其中$K$表示信号（属于Orlicz空间）范数本质上集中的紧集。同样地，我们为混合范数Orlicz空间中的信号证明了随机采样结果。作为抽象理论的应用，我们获得了加权平移不变空间和Orlicz调制空间的采样定理。

英文摘要

We study the sampling problem for non-decaying signals defined on a locally compact group. The signals are modeled as elements of suitable subspaces of weighted Orlicz and weighted mixed-norm Orlicz spaces, thereby allowing controlled growth at infinity. Our principal focus is on image spaces of idempotent integral operators. Under suitable oscillation estimates on the associated integral kernels, we establish deterministic sampling theorems that guarantee stable reconstruction from pointwise samples. We further prove an average sampling result in the same setting. In addition, we prove random sampling theorems for functions whose norm is essentially concentrated on a compact subset of the group. In particular, we show that, with high probability, sampling is possible from $\mathcal{O}\!\left(μ(K)\log μ(K)\right)$ randomly chosen sample points, where $K$ denotes the compact set on which the signal, belonging to Orlicz space, is essentially norm concentrated. In the same spirit, a random sampling result is proved for signals in mixed-norm Orlicz space. As applications of the abstract theory, we obtain sampling theorems for weighted shift-invariant spaces and Orlicz modulation spaces.

发表机构

• Indian Institute of Technology Delhi（印度德里理工学院）

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