On the Boundedness of Oscillatory Integral Operators in Harmonic Analysis

Posted May 12th, 2010 at 5:03 pm.

Manal Zaher

Mentor: Dr. Leslie Cheng

My summer research is in the area of harmonic analysis, the branch of mathematics which 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 this branch of mathematics, there is great concern about examining mathematical objects by decomposing them into many simpler building blocks and studying these simpler components. This would make the study of the original object easier and more comprehensible in most cases. For instance, to study a function (a musical sound, for example) we could break it down into a sum of the pure harmonics, sine and cosine, which will allow us to study these simple graphs, to better understand our original function’s behavior. Harmonic analysis has become a vast subject with applications in areas as diverse as acoustics, optics, electrical engineering, quantum mechanics, and neuroscience.

My research will focus on some theoretical aspects of the field – in particular the boundedness of oscillatory integral operators. I have started studying previous work in the field as a background. I will try to extend some of the results obtained by BMC undergraduates in the past (Zoryanna Dopko ’05, Kirsten Kemp ’07, Sarah Khasawinah ’09). Once I have finished learning the techniques used in their proofs, I will work on similar problems extending their results by modifying their techniques.

Filed under: 2009,Cheng, Dr. Leslie,Zaher, Manal by Ann Dixon

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