Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds

Sobolev, Besov and Paley-Wiener vectors in Banach and Hilbert spaces

We consider Banach spaces equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. Using these semigroups we introduce an analog of a modulus of continuity and define analogs of Besov norms. A generalization of a classical interpolation theorem is proven in which the role of Sobolev spaces is played by subspaces defined in terms of infinitesimal operators of these semigroups. We show that our assumptions about a given set of semigroups are satisfied in the case of a strongly continuous bounded representation of a Lie group. In the case of a unitary represent…

滤过李群上的Besov与Triebel-Lizorkin空间，II：逐点乘法、局部化与差分

Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, II: Pointwise Multiplication, Localization, Differences

本文在实连通李群上发展加权次强制算子相关的Besov与Triebel-Lizorkin空间理论，研究其代数性质、逐点乘子、局部化范数及差分刻画。

Localisation uniforme des espaces de Besov et de Lizorkin-Triebel

We give intrinsic characterisations for the uniformly localized versions of the Besov spaces $B^{s}_{p,q}({\mathbb R}^n)$, where $p,q\in [1,+\infty]$, and of the Lizorkin-Triebel spaces $F^{s}_{p,q}({\mathbb R}^n)$, where $q\in [1,+\infty]$ and $p\in [1,+\infty[$, whatever be the real number $s>0$.

与“ax+b”群相关的Besov空间和Paley-Wiener空间、连续模以及Hardy-Steklov算子

Besov and Paley-Wiener spaces, Moduli of continuity and Hardy-Steklov operators associated with the group $"ax+b"$

本研究针对直线仿射变换“ax+b”李群的三种表示，刻画了Sobolev、Besov与Paley-Wiener空间的关系，构建了L₂逼近理论并证明了Jackson型不等式。

Geometric Space-Frequency Analysis on Manifolds

This paper gives a survey of methods for the construction of space-frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace-Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections. After the discussion of the real line, our presentation starts o…

Bimodal Wilson systems in $L^2(\mathbb R)$

Given a window $ϕ\in L^2(\mathbb R),$ and lattice parameters $α, β>0,$ we introduce a bimodal Wilson system $\mathcal{W}(ϕ, α, β)$ consisting of linear combinations of at most two elements from an associated Gabor $\mathcal{G}(ϕ, α, β)$. For a class of window functions $ϕ,$ we show that the Gabor system $\mathcal{G}(ϕ, α, β)$ is a tight frame of redundancy $β^{-1}$ if and only if the Wilson system $\mathcal{W}(ϕ, α, β)$ is Parseval system for $L^2(\mathbb R).$ Examples of smooth rapidly decaying generators $ϕ$ are constructed. In addition, when $3\leq β^{-1}\in \mathbb N$, we prove that it is …

Phi-transform on domains

The goal of the present paper is to construct bandlimited highly localized and nearly tight frames on domains with smooth boundaries in Euclidean spaces. These frames are used do describe corresponding Besov spaces.

Redundancy for localized and Gabor frames

Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist, the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one. We will answer this question in the affirmative for $\ell^1$-localized frames. We then specialize our results to Gabor multi-frames with generators in $M^1(\R^d)$, and Gabor molecules with envelopes in $W(C,l^…

Heat kernel generated frames in the setting of Dirichlet spaces

Wavelet bases and frames consisting of band limited functions of nearly exponential localization on Rd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local sc…

关于通过热核的同质类型狄利克雷空间

On Dirichlet Spaces of Homogeneous Type Via Heat Kernel

本文提出了一种基于热核的同质类型狄利克雷空间框架，用于分析带限函数在不同空间中的表示和分解。

多项式曲线片在高维球面上的应用

Polynomial curvelets on higher-dimensional spheres

本文介绍了多项式曲线片，用于高维球面的多尺度分解和稀疏表示。主要贡献是推导出一个精确的局部化界，表明框架元素远离中心快速衰减，适用于位置基分析。

框架乘子与Toeplitz算子的提升性质

The Lifting Property for Frame Multipliers and Toeplitz Operators

本文证明正符号框架乘子是coorbit空间间的Banach同构，推广提升定理至Gabor框架和Fock空间，得到调制空间与加权Fock空间的新同构结果。

不可压缩Navier-Stokes方程系统的空间批判

Espaces critiques pour le système des equations de Navier-Stokes incompressibles

在这项工作中，我们展示了抽象的功能空间E上的条件，确保存在一个整体的温和的解决方案的小数据在E或存在的局部温和的解决方案，在没有大小限制的一类半线性抛物型方程，其中包含不可压缩的Navier-Stokes系统作为一个基本的例子。我们还给出了一个抽象的判据，对所获得的解决方案的正则性。这些条件，在Littlewood-Paley估计产品的频谱本地化元素的$E$，是简单的检查在所有已知的情况下：勒贝格，洛伦茨，Besov，Morrey.空间.这些条件也适用于非不变空间E，我们给出了一些2-微局部空间的情况下的全部细节。以下评论没有出现在第一版上：这篇文章写于1998-99年左右，从未发表过，因为当时Koch和Tataru在$BMO^{-1}$中公布了他们关于Navier-斯托克斯方程的适定性的结果。我们相信，虽然这里的一些结果和反例是独立的利益，我们使他们电子。