Sampling Discretization of Integral Norms and Its Application

The paper addresses a 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 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 structure, namely, subspaces with tensor product structure. We demonstrate that the application of theorems based on Nikol'skii-type inequalities provides somewhat better results than the application of theorems based on entropy numbers conditions. Finally, we apply discretization results to the problem of sampling recovery.