Interpolations of Bargmann Type Measures

  • 1 DEPARTMENT OF MATHEMATICS, AICHI UNIVERSITY OF EDUCATION KARIYA, 448-8542, JAPAN
  • 2 MATHEMATICAL INSTITUTE, UNIVERSITY OF WROCŁAW, PL. GRUNWALDZKI 2/4, 50-384 WROCŁAW, POLAND

Abstract

In this paper, we shall discuss Bargmann type measures on C for several classes of probability measures on R. The unified interpolation expressions include not only the classical Bargmann measure and its q-deformation, but also their t-deformations and dilations. As a special case, we get conditions on existence and an explicit form of the Bargmann representation for the free Meixner family of probability measures.

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  • [1] N. Asai, I. Kubo, H.-H. Kuo, Segal–Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures, Proc. Amer. Math. Soc. 131(3) (2003), 815–823.

  • [2] N. Asai, Integral transform and Segal–Bargmann representation associated to q- Charlier polynomials, in: Quantum Information IV, T. Hida and K. Saitô (eds.), World Scientific, 2002, 39–48.

  • [3] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, I, Comm. Pure Appl. Math. 14 (1961), 187–214.

  • [4] C. Berg, M. Thill, Rotation invariant moment problem, Acta Math. 167(3–4) (1991), 207–227.

  • [5] M. Bozejko, W. Bryc, On a class of free Levy laws related to a regression problem, J. Funct. Anal. 236(1) (2006), 59–77.

  • [6] M. Bozejko, B. Kümmerer, R. Speicher, q-Gaussian processes: Non-Commutative and classical aspects, Comm. Math. Phys. 185 (1997), 129–154.

  • [7] M. Bozejko, J. Wysoczanski, New examples of convolutions and non-commutative central limit theorem, Banach Center. Publ. 43 (1998), 95–103.

  • [8] M. Bozejko, J. Wysoczanski, Remarks on t-transformations of measures and convolutions, Ann. Inst. H. Poincare Probab. Statist. 37(6) (2001), 737–761.

  • [9] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978.

  • [10] A. Hora, N. Obata, Quantum Probability and Spectral Analysis of Graphs, Springer, 2007.

  • [11] A. D. Krystek, Ł. J. Wojakowski, Bargmann measures for t-deformed probability, Probab. Math. Statist. 34(2) (2014), 279–291.

  • [12] H. van Leeuwen, H. Maassen, A q deformation of the Gauss distribution, J. Math. Phys. 36 (1995), 4743–4756.

  • [13] B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), 82–203.

  • [14] F. H. Szafraniec, Operators of the q-oscillator, Banach Center Publ. 78 (2007), 293–307.

  • [15] Ł. J. Wojakowski, Probability interpolating between free and boolean, Dissertationes Math. 446(45) (2007), 45pp.

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