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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
Unitary Representations of Lie Groups with Reflection Symmetry
We consider the following class of unitary representations of some
(real) Lie group which has a matched pair of symmetries described as
follows: (i) Suppose has a period-2 automorphism , and that the
Hilbert space carries a unitary operator such that (i.e., selfsimilarity). (ii) An added symmetry is implied
if further contains a closed subspace having
a certain order-covariance property, and satisfying the -restricted positivity: , ,
where is the inner product in . From
(i)--(ii), we get an induced dual representation of an associated dual group
. All three properties, selfsimilarity, order-covariance, and positivity,
are satisfied in a natural context when is semisimple and hermitean; but
when is the -group, or the Heisenberg group, positivity is
incompatible with the other two axioms for the infinite-dimensional irreducible
representations. We describe a class of , containing the latter two, which
admits a classification of the possible spaces satisfying the axioms of selfsimilarity and order-covariance.Comment: 49 pages, LaTeX article style, 11pt size optio
Coorbit Spaces for Dual Pairs
In this paper we present an abstract framework for construction of Banach
spaces of distributions from group representations. This generalizes the theory
of coorbit spaces initiated by H.G. Feichtinger and K. Gr\"ochenig in the
1980's. Spaces that can be described by this new technique include the whole
Banach-scale of Bergman spaces on the unit disc. For these Bergman spaces we
show that atomic decompositions can be constructed through sampling. We further
present a wavelet characterization of Besov spaces on the forward light cone
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