Mentors: Professors Leslie Cheng and Rhonda Hughes
In general when we have smooth functions it is appropriate to use sine and cosine functions to analyze these functions (this is a method called the Fourier Series). When functions are not smooth (like those with abrupt changes) it is hard to use cosines and sines to measure these functions and in this case we turn to wavelets. We are able to lengthen or shorten the wavelets that we want in order to analyze these functions. Wavelets are defined to be mathematical functions which are used to divide certain functions into separate frequency components and then they are used to study each component. Applications of wavelets are often used in the analysis of data compression such as fingerprint compression by the FBI and signal analysis such as the analysis of music.
The goal of learning to use Wavelet Explorer is to create a manual so that students in the future will have a reference tool after they delve into the topics of Harmonic Analysis and Wavelets. After I have learned how the software works, I would like to apply it to heatlets to solve partial differential equation problems which are difficult to solve using other methods. I hope to create the manual using a program called LaTex which is used to easily write mathematical symbols.
Over the past year I have taken Professor Cheng’s Harmonic Analysis and Wavelet course and subsequently I began my own research in this area. Since there are many kinds of wavelets including Haar Wavelets, Daubechies Wavelets, Shannon Wavelets, etc. I hope to use the Wavelet Explorer to examine all of these different types of wavelets and their significant purposes for different situations. I will start by reading the literature about Wavelet Explorer that is located online that explains both about the theory behind each type of wavelet and the code used in the Wavelet Explorer. Next when the software becomes available, I hope to use it to solve difficult computational problems, those which I have been solving by hand. Finally, after becoming familiar with the program I hope to create a user friendly manual so that future students will have a guide to begin to use the program. Also at the end of the summer, Professor Hughes and I would like to use Wavelet Explorer to solve partial differential equation problems via heatlets.