# The Marcinkiewicz-Type Discretization Theorems

@article{Temlyakov2018TheMD, title={The Marcinkiewicz-Type Discretization Theorems}, author={Vladimir N. Temlyakov}, journal={Constructive Approximation}, year={2018}, volume={48}, pages={337-369} }

This paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. This problem is very important in applications, but there is no systematic study of it. We present here a new technique, which works well for discretization of the integral norm. It is a combination of probabilistic technique, based on chaining, and results on the entropy numbers in the uniform norm.

## 36 Citations

Entropy numbers and Marcinkiewicz-type discretization theorem

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- 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 Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials

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- 2017

The main goal of this paper is to study the discretization problem for the hyperbolic
cross trigonometric polynomials. This important problem turns out to be very
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- 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 of the Hyperbolic Cross Polynomials

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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.

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In this expository paper we will give a survey of some recent results concerning discretization of uniform and integral norms of polynomials and exponential sums which are based on various new…

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- Computer Science, MathematicsJ. Complex.
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Integral norm discretization and related problems

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The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above…

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.

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- 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…

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