# Sampling discretization and related problems

@article{Kashin2022SamplingDA,
title={Sampling discretization and related problems},
author={Boris Kashin and Egor D. Kosov and Irina Limonova and Vladimir N. Temlyakov},
journal={J. Complex.},
year={2022},
volume={71},
pages={101653}
}
• Published 15 September 2021
• Computer Science, Mathematics
• J. Complex.
9 Citations

### Some improved bounds in sampling discretization of integral norms

• Mathematics, Computer Science
ArXiv
• 2022
This paper proves sampling discretization results under a standard assumption formulated in terms of the Nikol’skii-type inequality, and obtains some upper bounds on the number of sam-ple points suﬃcient for good discretized of the integral L p norms of functions from ﬁnite-dimensional subspaces of continuous functions.

• Mathematics, Computer Science
Journal of Approximation Theory
• 2022

### On the reconstruction of functions from values at subsampled quadrature points

• Computer Science, Mathematics
• 2022
The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information, and regains the optimal rate since many of the lattice points are not needed.

### $\varepsilon$-isometric dimension reduction for incompressible subsets of $\ell_p$

Fix p ∈ [1,∞), K ∈ (0,∞) and a probability measure μ. We prove that for every n ∈N, ε ∈ (0,1) and x1, . . . ,xn ∈ Lp(μ) with ∥∥∥maxi∈{1,...,n} |xi |∥∥∥Lp(μ) ≤ K , there exists d ≤ 32e2(2K)2p logn ε2

### Stable phase retrieval in function spaces

• Mathematics
• 2022
Let (Ω,Σ, μ) be a measure space, and 1 ≤ p ≤ ∞. A subspace E ⊆ Lp(μ) is said to do stable phase retrieval (SPR) if there exists a constant C ≥ 1 such that for any f, g ∈ E we have (0.1) inf |λ|=1 ‖f

### Marcinkiewicz-Zygmund inequalities for polynomials in Fock space

• Mathematics
Mathematische Zeitschrift
• 2022
We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}

### Энтропия унитарного оператора на $L^2(\mathbb T^n)$

• Математический сборник
• 2022
В работе изучается понятие $\mu$-нормы оператора, введенное Д. В. Трещeвым. Мы концентрируемся на случае операторов на пространстве $L^2(\mathbb{T}^n)$, где $\mathbb{T}^n$ - $n$-мерный тор (случай

### High-Dimensional Geometric Streaming in Polynomial Space

• Computer Science
ArXiv
• 2022
The techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry, and yield nearly optimal trade-oﬀs between space and distortion for ℓ p subspace embeddings.

## References

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### Sampling Discretization of Integral Norms

• Mathematics, Computer Science
Constructive Approximation
• 2021
A conditional theorem for all integral norms Lq is obtained and a new Marcinkiewicz-type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses is derived.

### Integral norm discretization and related problems

• Mathematics
Russian Mathematical Surveys
• 2019
The problem is discussed of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. This problem is investigated

### Sampling Discretization of Integral Norms of the Hyperbolic Cross Polynomials

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.

### The Marcinkiewicz-Type Discretization Theorems

• V. Temlyakov
• Mathematics, Computer Science
Constructive Approximation
• 2018
A new technique is presented, which works well for discretization of the integral norm, which is a combination of probabilistic technique, based on chaining, and results on the entropy numbers in the uniform norm.

### A remark on discretization of the uniform norm

• Mathematics, Computer Science
ArXiv
• 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.

### A note on sampling recovery of multivariate functions in the uniform norm

• Mathematics
SIAM J. Numer. Anal.
• 2022
The recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm is studied to obtain preasymptotic estimates for the corresponding sampling numbers and a relation to the corresponding Kolmogorov numbers is pointed out.

### L2-norm sampling discretization and recovery of functions from RKHS with finite trace

• Mathematics, Computer Science
Sampling Theory, Signal Processing, and Data Analysis
• 2021
A spectral norm concentration inequality for infinite random matrices with independent rows for L_2-norm sampling discretization and recovery of functions in RKHS based on random function samples, where the kernel is assumed to be the finite trace of the kernel.

### Hyperbolic Cross Approximation

• Computer Science
• 2018
A survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions.