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 $\mathrm{SL}_2(\mathbb{R})$. 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 $\pi$ of some (real) Lie group $G$ which has a matched pair of symmetries described as follows: (i) Suppose $G$ has a period-2 automorphism $\tau$, and that the Hilbert space $\mathbf{H} (\pi)$ carries a unitary operator $J$ such that $J\pi =(\pi \circ \tau)J$ (i.e., selfsimilarity). (ii) An added symmetry is implied if $\mathbf{H} (\pi)$ further contains a closed subspace $\mathbf{K}_0$ having a certain order-covariance property, and satisfying the $\mathbf{K}_0$-restricted positivity: $\ge 0$, $\forall v\in \mathbf{K}_0$, where $$ is the inner product in $\mathbf{H} (\pi)$. From (i)--(ii), we get an induced dual representation of an associated dual group $G^c$. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context when $G$ is semisimple and hermitean; but when $G$ is the $(ax+b)$-group, or the Heisenberg group, positivity is incompatible with the other two axioms for the infinite-dimensional irreducible representations. We describe a class of $G$, containing the latter two, which admits a classification of the possible spaces $\mathbf{K}_0 \subset \mathbf{H} (\pi)$ 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