LSU Doctoral Dissertations

Identifier

etd-10272015-211853

Degree

Doctor of Philosophy (PhD)

Mathematics

Dissertation

Abstract

Banach spaces of functions, or more generally, of distributions are one of the main topics in analysis. In this thesis, we present an abstract framework for construction of invariant Banach function spaces from projective group representations. Coorbit theory gives a unified method to construct invariant Banach function spaces via representations of Lie groups. This theory was introduced by \Fch\, and \Gro\, in \cite{FG,FG1, FG2,FG3} and then extended in \cite{CO2}. We generalize this concept by constructing coorbit spaces using projective representation which is first studied by O. Christensen in \cite{O1}. This allows us to describe wider classes of function spaces as coorbits, in order to construct frames and atomic decompositions for these spaces. As in the general coorbit theory, we construct atomic decompositions and Banach frames for coorbit spaces under certain smoothness conditions. By this modification, we can discretize the Bergman spaces $A^p_{\alpha}(\B)$ via the family of projective representations $\{\rho_s\}$ of the group $\SU(n,1)$, for any real parameter $s>n$.

2015

Document Availability at the Time of Submission

Secure the entire work for patent and/or proprietary purposes for a period of one year. Student has submitted appropriate documentation which states: During this period the copyright owner also agrees not to exercise her/his ownership rights, including public use in works, without prior authorization from LSU. At the end of the one year period, either we or LSU may request an automatic extension for one additional year. At the end of the one year secure period (or its extension, if such is requested), the work will be released for access worldwide.

Olafsson, Gestur

COinS