数列空间中的限制性非线性逼近及其在小波基和插值中的应用

An Endpoint Version of Uniform Sobolev Inequalities

We prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [8]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [2]. We also prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.

Regular Families of Kernels for Nonlinear Approximation

This article studies sufficient conditions on families of approximating kernels which provide $N$--term approximation errors from an associated nonlinear approximation space which match the best known orders of $N$--term wavelet expansion. These conditions provide a framework which encompasses some notable approximation kernels including splines, so-called cardinal functions, and many radial basis functions such as the Gaussians and general multiquadrics. Examples of such kernels are given to justify the criteria, and some computational experiments are done to demonstrate the theoretical resul…

Spaces of Besov-Sobolev type and a problem on nonlinear approximation

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\uı spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation …

小波分解中的测度

Measures in wavelet decompositions

在应用中，希尔伯特空间H中规范正交基的选择可能来自于某些特定的交换算子代数的同时对角化。最近人们注意到，有一定的有限集的非交换运营商F_i，首先介绍了工程师在信号处理，这有助于澄清这种联系，并在同一时间抛出光分解的可能性小波包中使用的金字塔算法。虽然算子F_i最初用于信号的正交镜像滤波器，但最近的论文表明，它们在各种现代小波构造中是普遍存在的，特别是在从基库中选择小波包时。这些是在信号或数据压缩问题中选择具有最佳频率集中的基的构造。虽然由F_i-系统生成的代数A是非交换的，在C*-代数理论中称为“Cuntz代数”，但它的每个表示都包含一个标准极大交换子代数，即，子代数是Gelfand空间X的某个C（X）。A的一个给定表示，限制在C（X），自然地导出X上的一个投影值测度，并且H中的每个向量导出X上的一个标量值测度。证明了一类诱导纯量测度的结构定理。在应用中，X可以是单位区间，或者是一个康托尔集;或者它可以是一个仿射分形，甚至是一个更一般的迭代极限，涉及由共形映射组成的迭代函数系统。

受限离散傅里叶变换

The restricted discrete Fourier transform

构造与离散傅里叶变换可交换的周期三对角矩阵，其单特征空间张成支撑在[-a,a]且变换支撑也在同一区间的函数空间，从而给出该受限变换的正交特征基和傅里叶唯一性对插值公式。

A trace theorem for Besov functions in spaces of homogeneous type

The aim of this paper is to prove a trace theorem for Besov functions in the metric setting, generalizing a known result from A. Jonsson and H. Wallin in the Euclidean case. We show that the trace of a Besov space defined in a `big set' $X$ is another Besov space defined in the `small set' $F\subset X$. The proof is divided in three parts. First we see that Besov functions in $F$ are restrictions of functions of the same type (but greater regularity) in $X$, that is we prove an Extension theorem. Next, as an auxiliary result that can also be interesting on its own, we show that the interpolati…

流形上的多调和及相关核：插值与逼近

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

针对紧致连通C∞黎曼流形（包括球面和SO(3)），利用微分几何和李群技术，证明由特定偏微分算子基本解生成的核具有一致有界且代数或指数衰减的Lagrange函数，进而得到插值和L2最小化的Lebesgue常数一致有界，并建立了紧致黎曼流形区域上的零点引理。

单位球面上核空间中的局部化基

Localized bases for kernel spaces on the unit sphere

针对球面上由正定或条件正定核生成的逼近/插值空间，提出一种基于仅O((log N)^2)个核组合的易计算、空间局部化、小足迹且鲁棒的基，并证明其L_p稳定性和多项式衰减估计，实现最优L_p逼近阶。