Harmonic and functional analysis of wavelet and frame expansions

Marin Bownik的这项数学研究项目专注于多维小波和框架展开数学理论的调和和泛函分析方面的研究。该项目的主要研究方向之一是发展构造大类非各向同性膨胀的局部化良好正交小波的技术。一个密切相关的互补主题是研究与扩张相关的经典函数空间的非各向同性类似。该项目的另一个方向是建造具有所需属性的框架，例如具有规定的规范和框架操作器。这一研究方向与Schur-Horn定理的无限维推广密切相关。Marcin Bownik的这一数学研究项目探索了小波和框架展开的数学理论。近年来，小波和框架理论在许多学科中得到了广泛的应用，包括应用和计算的谐波分析、信号处理和数据压缩。小波是关键工具的一些众所周知的例子包括JPEG2000数字图像标准和用于数据存储的指纹压缩。本项目旨在进一步加深数学理论，为小波和框架的进一步应用奠定基础。

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.