Sampling discretization and related problems

  title={Sampling discretization and related problems},
  author={Boris Kashin and Egor D. Kosov and Irina Limonova and Vladimir N. Temlyakov},
  journal={J. Complex.},

Some improved bounds in sampling discretization of integral norms

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.

Randomized weakly admissible meshes

On the reconstruction of functions from values at subsampled quadrature points

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

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

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 $L^2(\mathbb T^n)$

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


  • 2022

High-Dimensional Geometric Streaming in Polynomial Space

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.



Sampling Discretization of Integral Norms

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

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

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.

Universal discretization

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

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

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

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.