Survey of the theory of Fourier transforms on R^n, H^2, and homogeneous spaces of semisimple Lie groups

Undergraduate Honors Thesis

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Citeable URL: https://scholar.colorado.edu/concern/undergraduate_honors_theses/js956g90v

Abstract

The Fourier transform on R^n has many useful properties that prove to be useful in studying major problems arising in analysis—such as those arising in the study of differential equations. One can also develop the Fourier transform for abelian or compact locally compact Hausdorff groups, which shares many of the same remarkable properties of the Fourier transform on R^n. Even further, the Fourier transform can be defined on homogeneous spaces X = G/K where G is a connected noncompact semisimple Lie group with finite center and K is a maximal compact subgroup. The purpose of this paper is to survey the major properties and features of the Fourier transforms on these spaces, at the level of a student familiar with real analysis. We also explore how one can define and study pseudo-differential operators on such homogeneous spaces X = G/K.

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