Wavelet frames, filters, and operator algebras

Packer将从算子理论和算子代数的角度研究小波框架和相关的滤波函数。 使用这种方法，帕克和她的合作者希望阐明框架理论和过滤函数以及相关的算子和算子代数。 她将尝试构造更一般的小波族，包括基于分形空间的小波，首先由D。Dutkay和P.Jorgensen，以及广义的遗传学。 算子理论和算子代数以前使用的包装与合作者L。Baggett，K.梅里尔和Jorgensen在他们最近关于从广义滤波器系统构造Parseval框架的论文中。 Packer打算建立在这些结果的基础上，并希望使用一种方法，由于N。拉森和我Raeburn等人最近描述了一种利用Hilbert空间和部分等距的有向系统构造多分辨分析的方法。 在这些项目中的关键连接成分是家庭的过滤器功能，这些功能也可以用来构建运营商提供表示可识别的C* 代数，如Cuntz代数和Toeplitz代数。Larsen，Raeburn，梅里尔，Baggett和Packer打算利用这种新方法（1）从Baggett和梅里尔的抽象重数函数构造广义框架小波，（2）给出Dutkay和Jorgensen的分形小波的另一种构造方法。 Packer还希望扩展与M. Rieffel到更一般的非交换环面。 这可能有有趣的应用到Gabor框架。帕克、她的学生J. D 'Andrea和她的同事K.梅里尔最近使用了一些这些方法的空间上建立的Sierpinski垫片分形。通过这样做，他们能够获得一些有趣的数字照片分析。 小波和框架理论在信号处理和数字成像的数据存储和检索中非常有用。 另一方面，算子代数在抽象数学中起着重要的作用，并提供了分析和几何之间的桥梁。算子代数和小波的抽象理论与其应用之间的联系是有趣的，众所周知，多年来，Cuntz代数及其推广已经出现在各种现实生活中。 Packer打算在她以前与合作者在上述方向上的工作的基础上，获得该领域其他工作者感兴趣的新结果，从而促进该领域的其他研究。 她希望既能访问她的国内和海外合作者，并让他们访问科罗拉多，从而受益于各种机构。 帕克有一个学生完成了博士学位。她的另一个学生对分形小波在图像压缩和数据存储中的应用感兴趣。 第三个学生对算子代数中更多的理论问题感兴趣。这一研究领域的理论和应用的混合为研究生带来了许多问题，帕克希望吸引新的博士。学生在这些领域和相关领域的工作分析。

Packer will study wavelet frames and the related filter functions from the point of view of operator theory and operator algebras. Using this approach, Packer and her collaborators hope to shed light on both frame theory and filter functions and the related operators and operator algebras. She will attempt to construct more general families of wavelets, including wavelets based on fractal spaces, first introduced by D. Dutkay and P. Jorgensen, and on generalized solenoids. Operator theory and operator algebras were previously used by Packer with collaborators L. Baggett, K. Merrill, and Jorgensen, in their recent papers on the construction of Parseval frames from generalized filter systems. Packer intends to build on these results, and in addition hopes to use an approach due to N. Larsen and I. Raeburn, who have recently described a method to constructing multiresolution analyses by using directed systems of Hilbert spaces and partial isometries. The key linking ingredient in these projects is the family of filter functions, and these functions can also be used to construct operators giving representations of identifiable C*-algebras such as the Cuntz algebra and the Toeplitz algebra. Larsen, Raeburn, Merrill, Baggett and Packer intend to use this new approach to (1) build generalized frame wavelets from the abstract multiplicity function of Baggett and Merrill, and (2) give another approach towards constructing the fractal wavelets of Dutkay and Jorgensen. Packer also hopes to extend the concept of projective multiresolution analyses developed with M. Rieffel to more general non-commutative tori. This could have intriguing applications to Gabor frames. Packer, her student J. D'Andrea, and her colleague K. Merrill have recently used some of these methods on the space built from the Sierpinski gasket fractal. In so doing they were able to obtain some interesting analysis of digital photographs. Wavelet and frame theory are extremely useful in data storage and retrieval, both for signal processing and digital imaging. On the other hand, operator algebras play an important role in abstract mathematics and provide a bridge between analysis and geometry. The connection between the abstract theory of operator algebras and wavelets and their applications is an intriguing one, and it is well-known that Cuntz algebras and their generalizations have shown up in a variety of real-life situations over the years. Packer intends to build on her previous work with collaborators in the directions outlined above to obtain new results of interest to other workers in the field, thereby stimulating other research in this area. She hopes both to visit her domestic and overseas collaborators and have them visit Colorado, thereby benefiting a variety of institutions. Packer has had one student complete a Ph.D. thesis on this topic, and another of her students is interested in the use of fractal wavelets in image compression and data storage. A third student is interested in more theoretical problems in operator algebras. The mixture of the theoretical and the applied in this area of research generates many problems for graduate students to pursue, and Packer hopes to attract new Ph.D. students to work in these areas and related areas of analysis.