Harmonic and functional analysis of wavelet and frame expansions

点击翻译按钮获取中文摘要

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.