Nikol'skii-type inequalities for rearrangement invariant spaces

关于Nikol'skii的一个定理

On a theorem of Nikol'skii

本文提出Bernstein惰性定理，并研究该定理与自反性之间的联系。

A Bourgain-Brezis-Mironescu characterization of higher order Besov-Nikol'skii spaces

We study a class of nonlocal functionals in the spirit of the recent characterization of the Sobolev spaces $W^{1,p}$ derived by Bourgain, Brezis and Mironescu. We show that it provides a common roof to the description of the $BV(\mathbb{R}^N)$, $W^{1,p}(\mathbb{R}^N)$, $B_{p,\infty}^s(\mathbb{R}^N)$ and $C^{0,1}(\mathbb{R}^N)$ scales and we obtain new equivalent characterizations for these spaces. We also establish a non-compactness result for sequences and new (non-)limiting embeddings between Lipschitz and Besov spaces which extend the previous known results.

High-dimensional nonlinear approximation by parametric manifolds in Hölder-Nikol'skii spaces of mixed smoothness

We study high-dimensional nonlinear approximation of functions in Hölder-Nikol'skii spaces $H^α_\infty(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$ having mixed smoothness, by parametric manifolds. The approximation error is measured in the $L_\infty$-norm. In this context, we explicitly constructed methods of nonlinear approximation, and give dimension-dependent estimates of the approximation error explicitly in dimension $d$ and number $N$ measuring computation complexity of the parametric manifold of approximants. For $d=2$, we derived a novel right asymptotic order of noncontinu…

Embedding theorems of function classes

In this paper we obtain the necessary and sufficient conditions for embedding results of different function classes. The main result is a criterion for embedding theorems for the so-called generalized Weyl-Nikol'skii class and the generalized Lipschitz class. To define the Weyl-Nikol'skii class, we use the concept of a $(λ,β)$-derivative, which is a generalization of the derivative in the sense of Weyl. As corollaries, we give estimates of norms and moduli of smoothness of transformed Fourier series.

若干函数类上积分范数下的稀疏采样恢复

Sparse sampling recovery in integral norms on some function classes

本研究在通用采样离散化、无条件性质及Nikol'skii型不等式假设下，证明基于$L_2$通用离散化采样点的弱正交匹配追踪算法可在$2\le p<\infty$的$L_p$范数下实现良好稀疏采样恢复，得到单函数Lebesgue型不等式与多元函数类误差界，丰富了采样恢复理论。