Operator Algebras, Representations, and Wavelets

算子代数、表示和小波变换E.T. JORGENSENJorgensen的提案研究将涉及几个数学领域，并将集中在应用、光谱和平铺对偶以及分形迭代过程。其基本方法来源于算子代数和表示理论，其应用有三个方面的联系：小波理论、算子以及离散域和连续域上运算之间的关系。小波理论关注的是将连续数据数字化以存储的数学工具，以及从存储数据中重建所需图像（或时间信号）的合成过程。所涉及的算法以滤波器组的名义进行，其惊人的效率部分源于分析数据中（隐藏的）自相似性的使用。观测值或时间信号是函数，函数类构成了空间。数值相关性给现有的空间，希尔伯特空间，增加了结构。在由离散数据导出的空间中有算子，在由连续信号导出的空间中有算子。前者适用于计算，而后者反映了现实世界。两者之间的操作者是Jorgensen研究的重点。由于符号是可编程的，离散域和连续域的运算之间的关系被作为符号来研究。将运算符赋给符号关系所涉及的数学称为表示理论。这三个领域的结合在数学和工程的界面上开辟了令人兴奋的新机会。该方案的重点是研究信号处理中高通/低通滤波器的“离散世界”和小波的“连续世界”之间的交织算子。在“分界”的两边都存在着重要的算子代数和表示理论问题，而交织算子则揭示了高维小波的核心问题。该提案描述了来自不同分析领域的工具，以及动力系统和算子代数，如何合并到小波分析的提议项目中。技术的多样性也是这门学科的魅力所在，它不断产生新的研究生活动。乔根森最近有几个学生完成了这方面的论文。

ABSTRACTDMS-9987777OPERATOR ALGEBRAS, REPRESENTATIONS AND WAVELETSPALLE E.T. JORGENSENJorgensen's proposal research will involve several areas of mathematics and will be focussed on applications, spectral and tiling duality, and fractal iteration processes. The basic methods derive from operator algebras and representation theory, and the connection to the applications is threefold: wavelet theory, operators, and relations between operations in the discrete and continuos domains. Wavelet theory concerns the mathematical tools involved in digitizing continuous data with view to storage, and the synthesis process, recreating the desired picture (or time signal) from the stored data. The algorithms involved go under the name of filter banks, and their spectacular efficiency derives in part from the use of (hidden) self-similarity in the data which is analyzed. Observations or time signals are functions, and classes of functions make up spaces. Numerical correlations add structure to the spaces at hand, Hilbert spaces. There are operators in the spaces deriving from the discrete data and others from the spaces of continuous signals. The first ones are good for computations, while the second reflect the real world. The operators between the two are the focus of Jorgensen's research.Relations between operations in the discrete and continuous domains are studied as symbols, because symbols are programmable. The mathematics involved in assigning operators to the symbolic relations is called representation theory. The combination of the three areas opens up exciting new opportunities at the interface of mathematics and engineering. A main point in the proposal is the study of intertwining operators between, on one side, the "discrete world" of high-pass/low-pass filters of signal processing , and on the other side, the "continuous world" of wavelets. There are significant operator-algebraic and representation-theoretic issues on both sides of the "divide", and the intertwining operators throw light on central issues for wavelets in higher dimensions. The proposal describes how the tool from diverse areas of analysis, as well as from dynamical systems and operator algebra, merge into the proposed project on wavelet analysis. The diversity of techniques is also a charm of the subject, which continues to generate new graduate student activity. Jorgensen had several recent students complete theses in the subject.