Recently, it has been discovered that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error being measured in the square norm. It was established that a simple greedy type algorithm -- Weak Orthogonal Matching Pursuit -- based on good points for universal discretization provides effective recovery in the square norm. In this paper we extend those results by replacing the square norm with other integral norms. In this case we need to conduct our analysis in a Banach space rather than in a Hilbert space, making the techniques more involved. In particular, we establish that a greedy type algorithm -- Weak Chebyshev Greedy Algorithm -- based on good points for the $L_p$-universal discretization provides good recovery in the $L_p$ norm for $2\le p<\infty$. Furthermore, we discuss the problem of stable recovery and demonstrate its close relationship with sampling discretization.

翻译：近期研究发现，平方范数的通用采样离散化结果在以平方范数度量误差的稀疏采样恢复中具有实用性。研究表明，基于通用离散化优质点的简单贪婪型算法——弱正交匹配追踪——能有效实现平方范数下的恢复。本文将平方范数替换为其他积分范数以扩展上述结果。在此情况下，我们需要在巴拿赫空间而非希尔伯特空间中进行理论分析，使得技术方法更为复杂。特别地，我们证明：基于$L_p$通用离散化优质点的贪婪型算法——弱切比雪夫贪婪算法——在$2\le p<\infty$条件下能有效实现$L_p$范数下的恢复。此外，本文还探讨了稳定恢复问题，并论证其与采样离散化之间的紧密关联。