Harmonic and functional analysis of wavelet and frame expansions

这个数学研究项目由Marcin Bownik重点研究了多维小波和框架展开的数学理论的谐波分析和泛函分析方面。该项目的主要研究方向之一是开发用于构造大类别非各向同性膨胀膨胀的良好定域正交小波的技术。一个密切相关的互补课题是与膨胀膨胀相关的经典函数空间的非各向同性类似物的研究。项目的另一个方向是构造具有规定规范和框架算子等期望属性的框架。这一研究方向与舒尔-霍恩定理的无限维推广密切相关。这个数学研究项目由Marcin Bownik探索小波和框架展开的数学理论。近年来，小波和框架理论在应用和计算谐波分析、信号处理和数据压缩等领域得到了广泛的应用。小波作为关键工具的一些著名例子包括JPEG 2000数字图像标准和用于数据存储的指纹压缩。本项目旨在进一步发展数学理论，为小波和帧的进一步应用提供基础。

This mathematics research project by Marcin Bownik is focused on the investigation of the harmonic - and functional-analytic aspects of the mathematical theory of multi- dimensional wavelet and frame expansions. One of the main research directions of the project is the development of techniques for the construction of well-localized orthogonal wavelets for large classes of non-isotropic expansive dilations. A closely related complementary topic is the study of non-isotropic analogues of classical function spaces associated to expansive dilations. Another direction of the project is the construction of frames with desired properties such as with prescribed norms and frame operator. This line of research is closely related with the infinite dimensional generalizations of the Schur- Horn theorem.This mathematics research project by Marcin Bownik explores the mathematical theory of wavelet and frame expansions. In recent years, wavelet and frame theory has found many applications to a wide range of disciplines including applied and computational harmonic analysis, signal processing and data compression. Some well -known examples where wavelets are a key tool include the JPEG 2000 digital image standard and fingerprint compression for data storage. This project aims to further the mathematical theory that provides the foundation for further applications of wavelets and frames.