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    Examples of Coorbit Spaces for Dual Pairs

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    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 SL2(R)\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

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    We consider the following class of unitary representations π\pi of some (real) Lie group GG which has a matched pair of symmetries described as follows: (i) Suppose GG has a period-2 automorphism τ\tau , and that the Hilbert space H(π)\mathbf{H} (\pi) carries a unitary operator JJ such that Jπ=(π∘τ)JJ\pi =(\pi \circ \tau)J (i.e., selfsimilarity). (ii) An added symmetry is implied if H(π)\mathbf{H} (\pi) further contains a closed subspace K0\mathbf{K}_0 having a certain order-covariance property, and satisfying the K0\mathbf{K}_0 -restricted positivity: ≥0 \ge 0, ∀v∈K0\forall v\in \mathbf{K}_0 , where is the inner product in H(π)\mathbf{H} (\pi). From (i)--(ii), we get an induced dual representation of an associated dual group GcG^c. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context when GG is semisimple and hermitean; but when GG is the (ax+b)(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 GG, containing the latter two, which admits a classification of the possible spaces K0⊂H(π)\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

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    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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