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My thesis explores the concept of generalizing multiresolution analyses defined over finite intervals to include biorthogonal scaling functions. The objective is to have natural and fast analogues of the discrete wavelet transform for data of finite length. There are already some constructions of such objects, and my dissertation explores one approach (via periodization).

One future goal is expanding the definition of biorthogonal multiresolution analyses outside of L^2, where these families ordinarily belong. My dissertation contains results regarding families of distributions dual to L^2 functions, which is curious theoretically and more useful in applications.

My oral exam and dissertation proposal contain some detailed information about my work.

MRA on a finite interval, consisting of Daubechies scaling functions.

I enjoy analysis in general - particularly approximation theory, measure theory, probability, statistics, functional analysis, Fourier analysis, and harmonic analysis. My dissertation involved wavelet theory.

In the longterm I am fascinated by tomography, various medical applications (e.g. medical imaging, biostatistics, modeling of health-related statistics, and algorithms related to medical devices), information/communication and coding theory, signal/image processing, pattern recognition, and inverse problems.