The paper addresses the problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under two standard kinds of assumptions -- conditions on the entropy numbers and conditions in terms of the Nikol'skii-type inequalities. We prove some upper bounds on the number of sample points sufficient for good discretization and show that these upper bounds are sharp in a certain sense. Then we apply our general conditional results to subspaces with special structures, namely, subspaces with the tensor product structure. We demonstrate that applications of results based on the Nikol'skii-type inequalities provide somewhat better results than applications of results based on the entropy numbers conditions. Finally, we apply discretization results to the problem of sampling recovery.

翻译：该文件探讨了对符合某些条件的有限维次空间各组成部分整体规范的抽样分解问题。我们证明在两种标准假设 -- -- 以Nikol'skii型的不平等为条件的酶数和条件的条件 -- -- 下抽样分解结果。我们证明在样本点数量上有一些上限,足以实现良好的分解,并表明这些上限在某种意义上是锐利的。然后,我们将我们的一般有条件结果应用到具有特殊结构的子空间,即带有沙藻产品结构的子空间。我们证明,基于Nikol'skii型不平等的结果的应用结果比基于酶号条件的结果应用结果的结果要好一些。最后,我们对取样的回收问题应用了分解结果。