Mentor: Dr. Leslie Cheng
My summer research in mathematics, conducted under the guidance of Professor Leslie Cheng, will consist of two main components: a book project on harmonic analysis and wavelets, and an investigation into oscillatory integral operators.
The first part initially involves gaining an understanding of harmonic analysis, a field of study that analyzes mathematical objects by decomposing them into more elementary components. Many functions, for example, can be decomposed into sums of cosine and sine; examining these relatively simple curves yields information about the original function. Harmonic analysis has wide applications not only in mathematics but also in medicine; radio, television, and music recording; spectral analysis; optics and fiber-optic communications; and telecommunications. Although it is a subject typically dependent on graduate-level material, it is one that Professor Cheng, together with Professor Rhonda Hughes, hopes to make accessible to undergraduates. To this end, they are compiling notes for an introductory book on harmonic analysis and wavelets geared toward students with only a calculus and linear algebra background. My work will consist of first reading, annotating, and editing these notes, and then typing them into the computer program LaTeX, which uses precise commands to generate publishable manuscripts.
The second part of my research also entails preliminary independent exploration, supplemented by occasional relevant seminars with my advisor. I will be reading graduate real analysis notes to obtain the appropriate background to read and investigate two related areas of interest: weighted Lp – L8 inequalities for oscillatory integral operators, and Lp estimates for oscillatory integral operators. There are two primary components to this study: Lp spaces and oscillatory integral operators. An Lp space is a special collection of measurable functions with Lp norms (defined by a particular integral) that are finite. An oscillatory integral operator takes an appropriate Lp function to a new function represented by the integral of the Lp function together with an oscillating factor (so defined since it involves sines and cosines) and a kernel (a function in two variables). Such operators are useful in solving heat and wave equations, in developing probability and physics theorems, and in processing signals, among many applications. Yet although geologists, biologists, physicists, economists, and specialists in other fields have widely recognized these applications of Lp functions and oscillatory integral operators, the properties of the functions themselves have been largely unexplored. This research, then, takes a theoretical approach, motivated by the well-documented applied approach, to investigating the properties of these functions, particularly their boundedness. In doing so, it intends to provide a basic theory of knowledge that explains the behavior of functions with applications across the sciences.