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arXiv · 2001.09320

Sampling discretization of integral norms

F. Dai·A. Prymak·A. Shadrin·V. Temlyakov·S. Tikhonov
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Abstract

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun recently. In this paper we obtain a conditional theorem for all integral norms $L_q$, $1\le q<\infty$, which is an extension of known results for $q=1$. To discretize the integral norms successfully, we introduce a new technique, which is a combination of probabilistic technique with results on the entropy numbers in the uniform norm. As an application of the general conditional theorem, we derive a new Marcinkiewicz type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses.

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BibTeXRIS

F. Dai, A. Prymak, A. Shadrin, V. Temlyakov, S. Tikhonov. 2020-01-25. Sampling discretization of integral norms. https://arxiv.org/abs/2001.09320

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