# Entropy numbers and Marcinkiewicz-type discretization theorem

@article{Dai2020EntropyNA, title={Entropy numbers and Marcinkiewicz-type discretization theorem}, author={Feng Dai and A. V. Prymak and Alexei Shadrin and Vladimir N. Temlyakov and Sergei Yur'evich Tikhonov}, journal={arXiv: Classical Analysis and ODEs}, year={2020} }

This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied to establish a Marcinkiewicz type discretization theorem for integral norms of functions from a given finite dimensional subspace.

## 13 Citations

Sampling Discretization of Integral Norms of the Hyperbolic Cross Polynomials

- Mathematics, Computer ScienceArXiv
- 2020

It is shown that recently developed techniques allow for a new Marcinkiewicz type discretization theorem for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses.

Sampling discretization of integral norms

- Mathematics, Computer ScienceArXiv
- 2020

A conditional theorem for all integral norms of functions from a given finite dimensional subspace is obtained, which is an extension of known results for q=1 and a new Marcinkiewicz type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses is derived.

Sampling discretization of integral norms and its application

- Computer Science, MathematicsArXiv
- 2021

This paper proves sampling discretization results under two standard kinds of assumptions – conditions on the entropy numbers and conditions in terms of the Nikol’skii-type inequalities, and applies these results to subspaces with special structure.

Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness

- Computer Science, MathematicsJ. Complex.
- 2021

It is demonstrated how the results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the L 2 norm of functions with small mixed smoothness.

Sampling discretization and related problems

- Mathematics
- 2021

This survey addresses sampling discretization and its connections with other areas of mathematics. We present here known results on sampling discretization of both integral norms and the uniform norm…

A remark on entropy numbers

- Mathematics, Computer ScienceArXiv
- 2020

Talagrand's fundamental result on the entropy numbers is slightly improved. Our proof uses different ideas based on results from greedy approximation.

FA ] 2 2 Se p 20 20 On sampling discretization in L 2

- 2020

A sampling discretization theorem for the square norm of functions from a finite dimensional subspace satisfying Nikol’skii’s inequality is proved. The obtained upper bound on the number of sampling…

On optimal recovery in L 2 .

- Mathematics, Computer Science
- 2020

We prove that the optimal error of recovery in the $L_2$ norm of functions from a class $\bF$ can be bounded above by the value of the Kolmogorov width of $\bF$ in the uniform norm. We demonstrate on…

Marcinkiewicz-type discretization of $L^p$-norms under the Nikolskii-type inequality assumption

- Mathematics
- 2020

The paper studies the Marcinkiewicz-type discretization problem for integral norms on subspaces of $L^p$. Certain close to optimal results are obtained on subspaces for which the Nikolskii-type…

A remark on discretization of the uniform norm

- Computer ScienceArXiv
- 2021

A general result is proved, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

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