Harmonic Analysis and Affinely Invariant Measures

Florida State University, Tallahassee FL

Investigators

Abstract

Abstract This project will focus on the relationship of affinely-invariant measures to certain problems in harmonic analysis. Drury observed that affine arclength appears to be the natural measure to use when considering averaging and Fourier restriction operators associated with certain curves. More recently, the PI has obtained certain uniform results along these lines for curves in two dimensions and for surfaces in higher dimensions. He intends to continue these investigations and also to consider some questions of an integral-geometric nature. The latter stem from the concept of affine dimension, an analog of Hausdorff dimension which also takes into account a set's curvature. The averaging operators considered in this project belong to a class of smoothing operators which play an important role in the study of physical processes like wave motion and heat propagation. These operators are also related to the Radon transforms which appear in tomography. Similar operators are important in signal analysis and communications theory. The Fourier restriction operators considered here are part of the theory of Fourier analysis. Their study contributes to the general understanding of the Fourier transform, a tool which is vital to applications of mathematics ranging from electrical engineering and communications theory through fluid dynamics.