Alternatives to the tensor product in wavelet construction and beyond

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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.