Operator Algebras, Representations, and Wavelets

算子代数、表示和小波空间JORGENSENJorgensen的提案研究将涉及数学的几个领域，并将集中在应用程序，光谱和平铺对偶，分形迭代过程。 基本方法来自算子代数和表示论，与应用的联系是三方面的：小波理论，算子，以及离散和连续域中操作之间的关系。 小波理论涉及的数学工具，涉及到数字化连续数据的存储，以及合成过程，从存储的数据中重新创建所需的图像（或时间信号）。 所涉及的算法以过滤器组的名义进行，其惊人的效率部分来自于在分析的数据中使用（隐藏的）自相似性。 观测或时间信号是函数，函数类构成空间。 数值相关性为手头的空间（希尔伯特空间）增加了结构。 在空间中有从离散数据导出的算子和从连续信号的空间导出的算子。 第一种方法适合于计算，而第二种方法反映了真实的世界。 两者之间的运算符是Jorgensen研究的重点，因为符号是可编程的，所以离散域和连续域中运算之间的关系被当作符号来研究。 将运算符分配给符号关系所涉及的数学称为表示论。 这三个领域的结合开辟了令人兴奋的数学和工程接口的新机会。 该提案的一个要点是研究交织算子之间的交织，一方面，高通/低通滤波器的信号处理的“离散世界”，另一方面，小波的“连续世界”。 有显着的运营商代数和代表性理论的问题，双方的“鸿沟”，交织运营商抛出光的中心问题，小波在更高的维度。 该提案描述了如何从不同的分析领域的工具，以及从动力系统和算子代数，合并到小波分析的拟议项目。 技术的多样性也是该学科的魅力，它继续产生新的研究生活动。 约根森最近有几个学生完成了这一主题的论文。

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.