Mentor: Professor Leslie Cheng
Harmonic Analysis examines mathematical objects by decomposing them into elementary components. Knowledge of these components is necessary for understanding ideas in certain fields of interest. Areas such as signal compression and image processing can be improved with the use of harmonic analysis. For instance, Fourier Transforms make very efficient numerical approximations, which are often used in X-ray tomography. Fourier Transforms break the function into sums of sine and cosine and are used to reconstruct a three-dimensional object from two-dimensional images that the X-rays produce. I will specifically be researching medical imaging and the influence Harmonic Analysis has had on the advancements in X-ray tomography. It is vital for mathematicians to make advances in this area so that improvements can be made in this field.
The second part of my summer research involves working on a book project dealing with Harmonic Analysis. Professor Cheng and Professor Hughes are currently writing a textbook on Harmonic Analysis, which is a topic usually studied at the graduate level. They hope to produce a textbook that is unique in that it can be used by undergraduates who have a background only in calculus and linear algebra. I will be reading and reviewing their work to ascertain that it is clear and comprehensible to the undergraduate student and will help make any necessary changes so it is understandable. Reading their work will provide me with a background in Harmonic Analysis and make sure that the explanations are clear, useful and profitable for their target audience. I will first learn the background material necessary to do research in this area and my research project will continue during the school year.