Estimates of Fourier Transforms and Applications

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The goal of the project is to investigate the behavior of Fourier transforms under various restrictions and to obtain sharper estimates of Fourier transforms and Periodizations. The research will be focused on two main objectives. The first one is to obtain new versions for the Uncertainty Principle in Harmonic Analysis. The Uncertainty Principle is a statement which says that a function and its Fourier transform can not be simultaneously concentrated on small sets. The famous Heisenberg Uncertainty Principle in Quantum Mechanics is one of the examples of the more general Uncertainty Principle in Harmonic Analysis. In particular, the investigator will study functions with Fourier transforms supported on sets with certain type of densities to get new versions of the Uncertainty Principle and apply the results for Partial Differential Equations and Signal Processing. The second objective is devoted to the relation between functions and their periodizations over integer lattices in higher dimensions. Periodizations are often used in Harmonic Analysis as a link between Fourier series and Fourier integrals. They arise in problems having periodic structure that is why they are an important tool in such applied sciences as Electrical Engineering, Signal Processing and Crystallography. In this proposal the investigator studies various properties of Fourier transforms. The Fourier transform is a major mathematical tool extensively used to represent, convert and recover digital data, information and signals in Signal Processing, Computer Science and Electrical Engineering, Crystallography and Tomography. The author proposes to further investigate properties of Fourier transformations which should lead to a deeper understanding of the behavior of Fourier transforms under various restrictions. The development of the proposed Uncertainty Principle in Harmonic Analysis will lead to tools to determine how much information is sufficient to recover signals and data. In return, this study of Fourier transforms will give new techniques not only in theoretical areas such as Mathematical Physics, in particular, Partial Differential Equations but also in applied sciences such as Image Processing, Electrical Engineering and Numerical Methods.