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Square Integrable and Holomorphic Kernels

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Coherent States, Wavelets, and Their Generalizations

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Abstract

This chapter is devoted to a systematic study of two special types of reproducing kernel Hilbert spaces, namely, those corresponding to square integrable kernels and to holomorphic kernels. The first case is illustrated by the construction of CS on the circle. This example suggests introducing action-angle variables, which are then used to extend the theory to a non-holomorphic set-up, namely, the so-called Gazeau–Klauder CS. The latter in turn lead to probabilistic considerations, that will be the focus of Chap. 11.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada

    Syed Twareque Ali

  2. Institut de Recherche en Mathématique et Physique (IRMP), Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Jean-Pierre Antoine

  3. Astroparticules et Cosmologie (APC, UMR 7164), Université Paris Diderot, Sorbonne Paris Cité Paris, France

    Jean-Pierre Gazeau

  4. Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, Brasil

    Jean-Pierre Gazeau

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  1. Syed Twareque Ali
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  2. Jean-Pierre Antoine
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  3. Jean-Pierre Gazeau
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Ali, S.T., Antoine, JP., Gazeau, JP. (2014). Square Integrable and Holomorphic Kernels. In: Coherent States, Wavelets, and Their Generalizations. Theoretical and Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8535-3_6

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  • DOI: https://doi.org/10.1007/978-1-4614-8535-3_6

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