Wavelets, Coorbit Theory, and Projective Representations
type: CreativeWork
description: 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 $s>n$.
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 $s>n$.
author: Darweesh, Amer Hasan
keywords: Lie group, Projective representation, Coorbit space, Banach frame, Atomic decompositions, Bergman spaces
date published: 2015-01-01T08:00:00Z
date modified: 2026-07-24T02:57:39Z
provider: Lousiana State University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Mathematics

PropertyValue
property ID: DOI
value: 10.31390/gradschool_dissertations.448