Fourier and Fourier-type Algebras of Lie Groups

University Of Delaware, Newark DE

Investigators

Abstract

Harmonic analysis is a fundamental branch of mathematics that addresses the following central problem: How can a function or signal be represented or approximated as a combination of basic waves? Such representations of signals are heavily used in today's technology for storage, transmission, and noise reduction of signals. Unfortunately, harmonic analytic methods perform poorly in various circumstances, as they are designed based on the assumption that the outcome of a series of actions is unchanged, regardless of the order in which the actions are performed. However, actions are often not interchangeable. For example, the result of performing consecutive rotations in three-dimensional (3D) space is highly sensitive to the order of rotations. Indeed, the real world is a very "non-commutative" universe. The space of 3D rotations is a simple example of a Lie group, which arise naturally in physics. This project will use techniques of functional analysis and Lie theory to advance the theory of non-commutative harmonic analysis for Lie groups. Fourier transforms and its analogues are the cornerstone of classical harmonic analysis. To generalize the concept of a Fourier transform to non-Abelian groups, the modern field of non-commutative harmonic analysis was initiated. The broad philosophy here is to employ group representations and the theory of operators on Hilbert spaces to capture the non-Abelian nature of a group. A major trend in non-commutative harmonic analysis is to investigate function algebras related to the Fourier analysis (and representation theory) of non-Abelian groups. The Fourier algebra, which is associated with the regular representation of the ambient group, is a fundamental example of such function algebras. This project investigates Banach algebraic behavior, in particular derivation theory and spectral theory, of the Fourier algebra and its weighted versions for various classes of locally compact (Lie) groups. The ultimate goal is to investigate the connections between the Lie structure of a group and the Banach algebraic properties of its Fourier algebra. This project is jointly funded by the Analysis Program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.