Entropy numbers and Marcinkiewicz-type discretization theorem

关于多元$C^α$域上的Bernstein型与Marcinkiewicz型不等式

On Bernstein- and Marcinkiewicz-type inequalities on multivariate $C^α$-domains

该研究证明了1≤α≤2的一般紧$C^α$域边界上$L^p$空间的新Bernstein型与Markov型不等式，并将其用于建立多项式$L^p$范数离散化的渐近最优Marcinkiewicz型不等式。

Sampling discretization of integral norms

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, w…

$C^2$-域上的$L^p$-Bernstein不等式及其在离散化中的应用

$L^p$-Bernstein inequalities on $C^2$-domains and applications to discretization

本文在一般紧致$C^2$-域上建立了与边界切向导数相关的$L^p$-Bernstein型不等式，并据此得到$L^p$范数离散化的Marcinkiewicz型不等式和正求积公式，且采样数达到多项式空间维数的最优量级。

Sampling discretization of integral norms and its application

The paper addresses the problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under two standard kinds of assumptions -- conditions on the entropy numbers and conditions in terms of the Nikol'skii-type inequalities. We prove some upper bounds on the number of sample points sufficient for good discretization and show that these upper bounds are sharp in a certain sense. Then we apply our general conditional results to subspaces with special structures, namely, subspaces with the tenso…

小乘积集上一致范数的无维数离散化

Dimension-free discretizations of the uniform norm by small product sets

本文提出高维解析多项式一致范数的无维数离散化方法，采样集基数由个体次数而非总次数决定，并应用于循环群 Bohnenblust-Hille 不等式，得到显式常数。

Universal sampling discretization

Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability space $(Ω, μ)$ spanned by a uniformly bounded Riesz basis $Φ_N$. Given an integer $1\leq v\leq N$ and an exponent $1\leq q\leq 2$, we obtain universal discretization for integral norms $L_q(Ω,μ)$ of functions from the collection of all subspaces of $X_N$ spanned by $v$ elements of $Φ_N$ with the number $m$ of required points satisfying $m\ll v(\log N)^2(\log v)^2$. This last bound on $m$ is much better than previously known bounds which are quadratic in $v$. Our proof uses a conditional theorem on universal sampling d…