Geodesic
Sampling Discretization of Integral Norms and Its Application
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 106-119

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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.
Keywords: sampling discretization, entropy numbers, Nikol'skii inequality, recovery. 
@article{TM_2022_319_a8,
     author = {F. Dai and V. N. Temlyakov},
     title = {Sampling {Discretization} of {Integral} {Norms} and {Its} {Application}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {106--119},
     publisher = {mathdoc},
     volume = {319},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_319_a8/}
}
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F. Dai; V. N. Temlyakov. Sampling Discretization of Integral Norms and Its Application. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 106-119. http://geodesic.mathdoc.fr/item/TM_2022_319_a8/