Wavelet approximation theory in higher dimensions: Foundations for a systematic comparison of diverse wavelet systems

高维小波逼近理论:各种小波系统系统比较的基础

基本信息

项目摘要

Recent years have witnessed an increasing amount of wavelet-type systems specifically taylored to efficiently approximate salient structures in multivariate data. Prominent examples are tensor product wavelets, curvelets and shearlets. Some of these constructions are based on the representation theory of locally compact groups; most of them are in fact based on the affine actions of certain matrix groups. The aim of the current project is to develop a unified wavelet approximation theory for a large class of group-theoretically defined wavelet systems. An essential tool for this purpose will be a class of Banach spaces, the coorbit spaces, and the closely related decomposition spaces. The chief objective of the project is to continue the systematic study of approximation-theoretic properties of continuous and discrete wavelet systems arising from the natural action of a semidirect product group, initiated in recent work of the applicant.The main challenge consists in the development of pertinent, explicitly verifiable criteria on the dilation group that should serve as a basis of such an analysis; here we expect the dual action of the dilation group to be of central importance. The chief objectives related to coorbit spaces are the following: We would like to extend the scale of coorbit spaces under considerations, e.g., to include quasi-Banach spaces, as well as Triebel-Lizorkin-type spaces. Furthermore, we would like to develop a systematic understanding of embedding results, in part for the comparison of coorbit spaces associated to different dilation groups, but also for the comparison to classical smoothness spaces (e.g. Besov spaces) and more recently defined, anisotropic analogs. Here we intend to use the decomposition space language as a common framework for generalized and group-related wavelet transforms. Part of the research will be concerned with the extension of the decomposition space framework, with the aim of allowing an easier transition between coorbit and decomposition space theory. Concurrently, we intend to develop criteria for the suitability of a dilation group for the analysis of local regularity. Here we will focus on local Hölder regularity, and on the characterization of the wavefront set.
近年来,越来越多的小波类系统被用来有效地逼近多变量数据中的显著结构。突出的例子是张量积小波、曲线小波和剪切波。其中一些构造是基于局部紧群的表示理论,其中大多数实际上是基于某些矩阵群的仿射作用。本项目的目的是为一大类群理论定义的小波系统发展一个统一的小波逼近理论。为了达到这个目的,一个基本的工具将是一类Banach空间,共轨空间,以及密切相关的分解空间。该项目的主要目标是继续系统地研究由半直积群的自然作用产生的连续和离散小波系统的逼近理论性质。主要的挑战在于关于伸缩群的相关的、明确可验证的准则的发展,它应该作为这种分析的基础;在这里,我们期望伸缩群的对偶作用是至关重要的。与共轨空间有关的主要目标如下:我们希望扩大正在考虑的共轨空间的规模,例如,包括准Banach空间以及Triebel-Lizorkin型空间。此外,我们希望对嵌入结果有一个系统的理解,部分是为了比较与不同膨胀群有关的共轨空间,也是为了比较经典的光滑空间(例如Besov空间)和最近定义的各向异性类似物。在这里,我们打算使用分解空间语言作为广义和群相关小波变换的公共框架。部分研究将涉及分解空间框架的扩展,目的是允许在共轨和分解空间理论之间更容易地过渡。同时,我们打算为扩张组的适宜性制定标准,以分析局部正则性。在这里,我们将集中讨论局部Hölder正则性,以及波阵面集合的特征。

项目成果

期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Wavelet Coorbit Spaces viewed as Decomposition Spaces
  • DOI:
    10.1016/j.jfa.2015.03.019
  • 发表时间:
    2014-04
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Hartmut Fuhr;F. Voigtlaender
  • 通讯作者:
    Hartmut Fuhr;F. Voigtlaender
Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups
  • DOI:
    10.1016/j.acha.2016.03.003
  • 发表时间:
    2014-07
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Hartmut Fuhr;R. R. Tousi-R.
  • 通讯作者:
    Hartmut Fuhr;R. R. Tousi-R.
Resolution of the Wavefront Set Using General Continuous Wavelet Transforms
使用一般连续小波变换的波前集分辨率
{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Professor Dr. Hartmut Führ其他文献

Professor Dr. Hartmut Führ的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('Professor Dr. Hartmut Führ', 18)}}的其他基金

Mechanisms of sound localization investigated with head-related transfer functions
用头部相关传递函数研究声音定位机制
  • 批准号:
    280063511
  • 财政年份:
    2015
  • 资助金额:
    --
  • 项目类别:
    Research Grants
Effiziente Modellierung geometrischer Strukturen in digitalen Bildern Entwurf und Analyse neuer Algorithmen
数字图像中几何结构的有效建模新算法的设计和分析。
  • 批准号:
    18448685
  • 财政年份:
    2006
  • 资助金额:
    --
  • 项目类别:
    Research Grants
Interpretation und Implementation kontinuierlicher Wavelettransformationen von Funktionen mehrerer Veränderlicher, mit Bezug auf Anwendungen in der Bildverarbeitung
多变量函数连续小波变换的解释与实现,参考图像处理中的应用
  • 批准号:
    5184592
  • 财政年份:
    1999
  • 资助金额:
    --
  • 项目类别:
    Research Fellowships

相似国自然基金

非牛顿流方程(组)及其随机模型无穷维动力系统的研究
  • 批准号:
    11126160
  • 批准年份:
    2011
  • 资助金额:
    3.0 万元
  • 项目类别:
    数学天元基金项目
枢纽港选址及相关问题的算法设计
  • 批准号:
    71001062
  • 批准年份:
    2010
  • 资助金额:
    17.6 万元
  • 项目类别:
    青年科学基金项目

相似海外基金

Approximation theory of structured neural networks
结构化神经网络的逼近理论
  • 批准号:
    DP240101919
  • 财政年份:
    2024
  • 资助金额:
    --
  • 项目类别:
    Discovery Projects
Development of a novel best approximation theory with applications
开发一种新颖的最佳逼近理论及其应用
  • 批准号:
    DP230102079
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Discovery Projects
Conference: International Conference on Approximation Theory and Beyond
会议:近似理论及其超越国际会议
  • 批准号:
    2314578
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Control Theory, Qualitative Analysis, and Approximation of Coupled Structure-Flow Interaction Systems
耦合结构-流相互作用系统的控制理论、定性分析和逼近
  • 批准号:
    2348312
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Approximation Theory and Complex Dynamics
逼近理论和复杂动力学
  • 批准号:
    2246876
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Diophantine approximation and transcendental number theory
丢番图近似和超越数论
  • 批准号:
    RGPIN-2019-05618
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Discovery Grants Program - Individual
Approximation and coding theory techniques for distributed machine learning
分布式机器学习的近似和编码理论技术
  • 批准号:
    559010-2021
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Alexander Graham Bell Canada Graduate Scholarships - Doctoral
Approximation Theory and Elementary Submodels
近似理论和基本子模型
  • 批准号:
    2154141
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Control Theory, Qualitative Analysis, and Approximation of Coupled Structure-Flow Interaction Systems
耦合结构-流相互作用系统的控制理论、定性分析和逼近
  • 批准号:
    2206200
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Approximation and coding theory techniques for distributed machine learning
分布式机器学习的近似和编码理论技术
  • 批准号:
    559010-2021
  • 财政年份:
    2021
  • 资助金额:
    --
  • 项目类别:
    Alexander Graham Bell Canada Graduate Scholarships - Doctoral
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了