Stability of random sampling for shift-invariant signals in mixed Lebesgue spaces

Sampling and reconstruction are two important problems in sampling theory. In the case that the set of sampling satisfies the stability, the signal can be stably reconstructed from the discrete sampling set. For the purpose of signal reconstruction, the random sampling stability of shift-invariant signals in mixed Lebesgue spaces is studied, with emphasis on sampling stability of signals on energy concentration subsets. Under the condition that the generator satisfies the support and shift stability, the standardized subset of the energy concentration set is defined firstly. Secondly, we study the relationship between the infinite norm and the (p, q) norm of the signals on the energy concentration subsets and estimate the covering numbers of standardized subsets limited to a cube. Thirdly, the new sequence of random variables is defined according to the random sampling points uniformly distributed on the cube and its related properties are studied. Finally, the sampling inequality of energy concentrated signals is established by means of related lemma of covering numbers、corresponding property of random variables and Bernstein inequality. The results show that when there are enough sampling points, the sampling stability of the energy concentrated signals is established with high probability.