Abstract

Many problems in physics require the crafting of suitable tools for the analysis of data emanating from various non-Euclidean manifolds. The main tools, currently employed for this purpose, are Gabor type frames or general frames, and wavelets. Given this backdrop, the primary objective of this thesis is the development of wavelet-like and time frequency type transforms on certain non-Euclidean manifolds. An immediate example of such a manifold (in the sense that it is homeomorphic to several other two-dimensional manifolds of revolution) is the two-dimensional infinite cylinder, for which we construct here Gabor type frames and wavelets. The two-dimensional cylinder, as a surface of revolution, is naturally homeomorphic to several other two-dimensional manifolds (themselves also surfaces of revolution). Examples are the one-sheeted hyperboloid, the paraboloid with its apex removed, the sphere with two points removed, the ellipsoid with two points removed, the plane with the origin removed, the upper sheet, of the two sheeted hyperboloid, with one point removed, and so on. Using this fact, in this thesis we build Gabor type frames and wavelets on these manifolds. We also present a method for constructing wavelet-like transforms on a large class of such surfaces of revolution using a group theoretic approach. Finally, as a beginning to a related but different sort of study, we construct some localization operators associated to group representations, using symbols (in the sense of pseudo-differential operators) which are operator valued functions on the group.