Alternatives to the tensor product in wavelet construction and beyond

主要研究方向为从一维小波函数构造多维小波函数的张量积替代方法的研究。这些替代方法可以克服张量积方法的一些局限性，同时也补充了张量积公式。张量积概念在许多其他应用领域也很流行，因此新的发展有可能产生更广泛的影响。从数学的角度来看，研究包括扩展当前在开发余集和方法方面的努力，寻找构建小波基的替代机制，同时评估它们在某些应用中的区别性质，以及研究构建张量积不起作用的多维小波系统的系统方法。小波已经在包括图像压缩在内的广泛应用中得到了应用。小波作为关键工具的例子包括JPEG 2000数字图像标准和用于数据存储的指纹压缩。本工作围绕具体应用领域对多维小波系统的构建进行改进，为数学研究生从应用的角度研究数学提供了一个机会。

The principal investigator's research is focused on the study of alternatives to the tensor product for constructing multi-dimensional wavelet functions from one-dimensional wavelet functions. These alternative methods may overcome some limitations of the tensor product approach while also complementing the tensor product formulation. The tensor product concept is also prevalent in many other application areas, and thus new developments have the potential for broader impact. From the mathematical perspective, the research includes extending current efforts in developing the coset sum methodology, searching for alternative mechanisms to construct wavelet bases while assessing their distinguishing properties for certain applications, and studying systematic ways to construct multi-dimensional wavelet systems where the tensor product does not work. Wavelets have been used in a wide range of applications including Image Compression. Examples where wavelets are a key tool include the JPEG 2000 digital image standard and fingerprint compression for data storage. This work concerns improvements in the construction of multi-dimensional wavelet systems focused toward specific application areas, and provides an opportunity for mathematics graduate students to study mathematics from an application perspective.