Mentor: Professor Leslie Cheng
The field of harmonic analysis is the branch of mathematics that studies the representation of signals or functions as the superposition of basic waves. The basic waves are called “harmonics”, hence the name “harmonic analysis.” In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience.
Harmonic analysis has a long and rich history dating back to eighteenth century studies of the wave equation pioneered by Fourier. The succeeding 150 years saw a blossoming of Fourier analysis into a powerful set of tools, including the Fourier transform, in mathematical physics, engineering, applied partial differential equations, and pure mathematics. Because of its efficient numerical approximations, the Fourier transform forms the foundation for most image and signal processing algorithms. The 1930’s and 1940’s were a relatively quiet time for Fourier analysis, but, beginning in the 1950’s, the focus of Fourier analysis became singular integrals pioneered by Calderón and Zygmund. Singular integrals are mathematical objects that look infinite but when properly interpreted are finite and well behaved. These integrals form an important part of mathematical modeling of physical phenomena, such as heat conduction and wave motion; therefore, gaining understanding and establishing control of these integrals is of great importance.
One of the major events in harmonic analysis in the 1980’s was the development of excellent tools in data compression called wavelets by Yves Meyer in 1985. The new theory of wavelets has given harmonic analysis a way to reinvent itself. Now we can design a Fourier analysis to fit a given problem. The image compression technique currently used by the U.S. Federal Bureau of Investigation to electronically store fingerprints is based on a wavelet algorithm. The compression ratios are on the order of 20:1, and only experts can tell the difference between the original image and the decompressed image. There are many more applications of wavelets. Current methods for restoring recordings of Johannes Brahms (playing his First Hungarian Dance on the piano, for example) and of Italian opera singer Enrico Caruso use wavelet signal processing algorithms.
Harmonic analysis is usually taught at the graduate level. However, Professor Cheng and Professor Hughes are in the process of designing a textbook that enables undergraduates with only a linear algebra and calculus background, to be introduced to the concepts in harmonic analysis. I will be spending the summer helping them compile and type materials they need for their textbook. Last summer, I read the draft of the textbook and made comments. This summer I will address the comments I made and continue to help improve the textbook.