Square Integrable and Holomorphic Kernels

  • Syed Twareque Ali
  • Jean-Pierre Antoine
  • Jean-Pierre Gazeau
Part of the Graduate Texts in Contemporary Physics book series (GTCP)


In this chapter, we study two special types of reproducing kernel Hilbert spaces, which are probably the most widely occurring types in the physical literature. While the very general reproducing kernel Hilbert spaces, constructed in the last chapter, were spaces of vector-valued functions, they were not assumed to be Hilbert spaces of square integrable functions, with respect to any measure. Most reproducing kernel Hilbert spaces that arise in physics and in group representation theory do, on the other hand, turn out to be spaces of square integrable fuctions. Another widely occurring variety of reproducing kernel Hilbert spaces are spaces of holomorphic or square integrable holomorphic functions. We look at these two situations more closely in this chapter. Recall from the discussion in Chapter 2 that the family of canonical CS arise from a reproducing kernel Hilbert space of square integrable functions, and, indeed, they may also be associated to a space of analytic functions (the Bargmann space).


Hilbert Space Holomorphic Function Coherent State Moment Problem Integrable Kernel 
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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Syed Twareque Ali
    • 1
  • Jean-Pierre Antoine
    • 2
  • Jean-Pierre Gazeau
    • 3
  1. 1.Department of Mathematics and StatisticsConcordia University Loyola CampusMontréalCanada
  2. 2.Institut de Physique ThéoriqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Laboratoire Physique Théoretique and Matière CondenséeUniversité de Paris 7 — Denis DiderotParisFrance

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