389 research outputs found
Coorbits for projective representations with an application to Bergman spaces
Recently representation theory has been used to provide atomic decompositions
for a large collection of classical Banach spaces. In this paper we extend the
techniques to also include projective representations. As our main application
we obtain atomic decompositions of Bergman spaces on the unit ball through the
holomorphic discrete series for the group of isometries of the ball
Examples of Coorbit Spaces for Dual Pairs
In this paper we summarize and give examples of a generalization of the
coorbit space theory initiated in the 1980's by H.G. Feichtinger and K.H.
Gr\"ochenig. Coorbit theory has been a powerful tool in characterizing Banach
spaces of distributions with the use of integrable representations of locally
compact groups. Examples are a wavelet characterization of the Besov spaces and
a characterization of some Bergman spaces by the discrete series representation
of . We present examples of Banach spaces which
could not be covered by the previous theory, and we also provide atomic
decompositions for an example related to a non-integrable representation
Continuous Frames, Function Spaces, and the Discretization Problem
A continuous frame is a family of vectors in a Hilbert space which allows
reproductions of arbitrary elements by continuous superpositions. Associated to
a given continuous frame we construct certain Banach spaces. Many classical
function spaces can be identified as such spaces. We provide a general method
to derive Banach frames and atomic decompositions for these Banach spaces by
sampling the continuous frame. This is done by generalizing the coorbit space
theory developed by Feichtinger and Groechenig. As an important tool the
concept of localization of frames is extended to continuous frames. As a
byproduct we give a partial answer to the question raised by Ali, Antoine and
Gazeau whether any continuous frame admits a corresponding discrete realization
generated by sampling.Comment: 44 page
Wavelets, Coorbit Theory, and Projective Representations
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 of the group \SU(n,1), for any real parameter
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