We consider multi-variate signals spanned by the integer shifts of a set of generating functions with distinct frequency profiles and the problem of reconstructing them from samples taken on a random periodic set. We show that such a sampling strategy succeeds with high probability provided that the density of the sampling pattern exceeds the number of frequency profiles by a logarithmic factor. The signal model includes bandlimited functions with multi-band spectra. While in this well-studied setting delicate constructions provide sampling strategies that meet the information theoretic benchmark of Shannon and Landau, the sampling pattern that we consider provides, at the price of a logarithmic oversampling factor, a simple alternative that is accompanied by favorable a priori stability margins (snug frames). More generally, we also treat bandlimited functions with arbitrary compact spectra, and different measures of its complexity and approximation rates by integer tiles. At the technical level, we elaborate on recent work on relevant sampling, with the key difference that the reconstruction guarantees that we provide hold uniformly for all signals, rather than for a subset of well-concentrated ones. This is achieved by methods of concentration of measure formulated on the Zak domain.

翻译：我们考虑由一组具有不同频率分布特征的生成函数的整数平移张成的多变量信号，以及从随机周期集上的采样点重建信号的问题。我们证明，只要采样模式的密度超过频率分布数量的对数因子，这种采样策略就能以高概率成功。该信号模型包括多频带带限函数。尽管在这一经过充分研究的设定中，精细的构造提供了满足香农和兰道信息理论基准的采样策略，但我们所考虑的采样模式以对数过采样因子为代价，提供了一种简单替代方案，并具有有利的先验稳定裕度（紧凑框架）。更广泛地，我们还处理具有任意紧致谱的带限函数，以及通过整数平铺对其复杂性和逼近率的不同度量。在技术层面，我们详细阐述了关于相关采样的近期工作，关键区别在于我们提供的重建保证对所有信号一致成立，而非仅针对良好集中信号的子集。这通过在Zak域上制定的测度集中方法得以实现。