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An Approach to a Generalization of White Noise Analysis

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Modern Analysis and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 190))

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Abstract

In this article, we review some recent developments in white noise analysis and its generalizations. In particular, we describe the main idea of the biorthogonal approach to a generalization of white noise analysis, connected with the theory of hypergroups.

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References

  1. S. Albeverio, Yu.M. Berezansky, V. Tesko, A generalization of an extended stochastic integral. Ukrainian Math. J. 59 (2007), no. 5, 588–617.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Albeverio, Yu.L. Daletsky, Yu.G. Kondratiev, and L. Streit, Non-Gaussian infinite-dimensional analysis. J. Funct. Anal. 138 (1996), no. 2, 311–350.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Albeverio, Yu.G. Kondratiev, and L. Streit, How to generalize white noise analysis to non-Gaussian measures. Proc. Symp. Dynamics of Complex and Irregular Systems (Bielefeld, 1991); Bielefeld Encount. Math. Phys., VIII, World Sci. Publ., River Edge, NJ, 1993, pp. 120–130.

    Google Scholar 

  4. Yu.M. Berezansky, Spectral approach to white noise analysis. Proc. Symp. Dynamics of Complex and Irregular Systems (Bielefeld, 1991); Bielefeld Encount. Math. Phys., VIII, World Sci. Publ., River Edge, NJ, 1993, pp. 131–140.

    Google Scholar 

  5. Yu.M. Berezansky, Infinite-dimensional non-Gaussian analysis connected with generalized translation operators. Analysis on infinite-dimensional Lie groups and algebras (Marseille, 1997), World Sci. Publ., River Edge, NJ, 1998, pp. 22–46.

    Google Scholar 

  6. Yu.M. Berezansky, Commutative Jacobi fields in Fock space. Integr. Equ. Oper. Theory. 30 (1998), no. 2, 163–190.

    Article  MATH  MathSciNet  Google Scholar 

  7. Yu.M. Berezansky, Direct and inverse spectral problems for a Jacobi field. St. Petersburg Math. J. 9 (1998), no. 6, 1053–1071.

    MathSciNet  Google Scholar 

  8. Yu.M. Berezansky, On the theory of commutative Jacobi fields. Methods Funct. Anal. Topology 4 (1998), no. 1, 1–31.

    MathSciNet  Google Scholar 

  9. Yu.M. Berezansky, Spectral theory of commutative Jacobi fields: direct and inverse problems. Fields Inst. Conmun. 25 (2000), 211–224.

    MathSciNet  Google Scholar 

  10. Yu.M. Berezansky, Poisson measure as the spectral measure of Jacobi field. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), no. 1, 121–139.

    Article  MATH  MathSciNet  Google Scholar 

  11. Yu.M. Berezansky, Pascal measure on generalized functions and the corresponding generalized Meixner polynomials. Methods Funct. Anal. Topology 8 (2002), no. 1, 1–13.

    MathSciNet  Google Scholar 

  12. Yu.M. Berezansky and Yu.G. Kondratiev, Spectral Methods in Infinite-dimensional Analysis. Vols. 1, 2, Kluwer Academic Publishers, Dordrecht-Boston-London, 1995. (Russian edition: Naukova Dumka, Kiev, 1988).

    Google Scholar 

  13. Yu.M. Berezansky and Yu.G. Kondratiev, Non-Gaussian analysis and hypergroups. Funct. Anal. Appl. 29 (1995), no. 3, 188–191.

    Article  MATH  MathSciNet  Google Scholar 

  14. Yu.M. Berezansky and Yu.G. Kondratiev, Biorthogonal systems in hypergroups: an extension of non-Gaussian analysis. Methods Funct. Anal. Topology 2 (1996), no. 2, 1–50.

    MATH  MathSciNet  Google Scholar 

  15. Yu.M. Berezansky, V.O. Livinsky, and E.V. Lytvynov, A generalization of Gaussian white noise analysis. Methods Funct. Anal. Topology 1 (1995), no. 1, 28–55.

    MATH  MathSciNet  Google Scholar 

  16. Yu.M. Berezansky, E. Lytvynov, and D.A. Mierzejewski, The Jacobi field of a Levy process. Ukrainian Math. J. 55 (2003), no. 5, 853–858.

    Article  MathSciNet  Google Scholar 

  17. Yu.M. Berezansky and D.A. Mierzejewski, The chaotic decomposition for the Gamma field. Funct. Anal. Appl. 35 (2001), no. 4, 305–308.

    Article  MATH  MathSciNet  Google Scholar 

  18. Yu.M. Berezansky and D.A. Mierzejewski, The Construction of the Chaotic Representation for the Gamma Field. Infin. Diniens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 1, 33–56.

    Article  MATH  MathSciNet  Google Scholar 

  19. Yu.M. Berezansky and Yu.S. Samoilenko, Nuclear spaces of functions of an infinite number of variables. Ukrainian Math. J. 25 (1973), no. 6, 723–737 (in Russian).

    Google Scholar 

  20. Yu.M. Berezansky and V.A. Tesko, Spaces of fundamental and generalized functions associated with generalized translation. Ukrainian Math. J. 55 (2003), no. 12, 1907–1979.

    Article  MathSciNet  Google Scholar 

  21. Yu.M. Berezansky and V.A. Tesko, An orthogonal approach to the construction of the theory of generalized functions of an infinite number of variables, and Poissonian white noise analysis. Ukrainian Math. J. 56 (2004), no. 12, 1885–1914.

    Article  MathSciNet  Google Scholar 

  22. Yu.L. Daletsky, A biorthogonal analogy of the Hermite polynomials and the inversion of the Fourier transform with respect to a non-Gaussian measure. Funct. Anal. Appl. 25 (1991), no. 2, 138–40.

    Article  MathSciNet  Google Scholar 

  23. R.J. Elliott and J. van der Hoek, A general fractional white noise theory and applications to finance. Math. Finance 13 (2003), no. 2, 301–330.

    Article  MATH  MathSciNet  Google Scholar 

  24. T. Hida, Analysis of Brownian functionals. Carleton Math. Lect. Notes, 13 (1975).

    Google Scholar 

  25. T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise. An Infinite Dimensional Calculus. Kluwer Academic Publishers, Dordrecht, 1993.

    MATH  Google Scholar 

  26. H. Holden, B. Øksendal, J. Ubøe, and T. Zhang, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Birkhäuser, Boston, 1996.

    MATH  Google Scholar 

  27. K. Ito, Multiple Wiener Integral. J. Math. Soc. Japan 3 (1951), 157–169.

    Article  MATH  MathSciNet  Google Scholar 

  28. Y. Ito and I. Kubo, Calculus on Gaussian and Poissonian white noise. Nagoya Math. J. 111 (1988), 41–84.

    MATH  MathSciNet  Google Scholar 

  29. N.A. Kachanovsky, Biorthogonal Appell-like systems in a Hilbert space. Methods Funct. Anal. Topology 2 (1996), no. 3–4, 36–52.

    MathSciNet  Google Scholar 

  30. N.A. Kachanovsky and S.V. Koshkin, Minimality of Appell-like systems and embeddings of test function spaces in a generalization of white noise analysis. Methods Funct. Anal. Topology 5 (1999), no. 3, 13–25.

    MATH  MathSciNet  Google Scholar 

  31. Yu.G. Kondratiev, T. Kuna, M.J. Oliveira, Analytic aspects of Poissonian white noise analysis. Methods Funct. Anal. Topology 8 (2002), no. 4, 15–48.

    MATH  MathSciNet  Google Scholar 

  32. Yu.G. Kondratiev and E.W. Lytvynov, Operators of Gamma white noise calculus. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), no. 3, 303–335.

    Article  MATH  MathSciNet  Google Scholar 

  33. Yu.G. Kondratiev, J.L. Da Silva, and L. Streit, Generalized Appell systems. Methods Funct. Anal. Topology 3 (1997), no. 3, 28–61.

    MATH  MathSciNet  Google Scholar 

  34. Yu.G. Kondratiev, J.L. Da Silva, L. Streit, and G.F. Us, Analysis on Poisson and Gamma spaces. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), no. 1, 91–117.

    Article  MATH  MathSciNet  Google Scholar 

  35. Yu.G. Kondratiev, L. Streit, W. Westerkamp and J. Yan, Generalized functions in infinite-dimensional analysis. Hiroshima Math. J. 28 (1998), no. 2, 213–260.

    MATH  MathSciNet  Google Scholar 

  36. V.D. Koshmanenko and Yu.S. Samoilenko, The isomorphism between Fock space and a space of functions of infinitely many variables. Ukrainian Math. J. 27 (1975), no. 5, 669–674 (in Russian).

    MATH  Google Scholar 

  37. M.G. Krein, Fundamental aspects of the representation theory of Hermitian operators with deficiency index (m,m). Ukrainian Math. J. 1 (1949), no. 2, 3–66 (in Russian).

    MathSciNet  Google Scholar 

  38. M.G. Krein, Infinite J-matrices and a matrix moment problem. Dokl. Acad. Nauk SSSR 69 (1949), no. 2, 125–128 (in Russian).

    MATH  MathSciNet  Google Scholar 

  39. H.-H. Kuo, White noise distribution theory. CRC Press, 1996.

    Google Scholar 

  40. H.-H. Kuo, White noise theory. Handbook of Stochastic Analysis and Applications D. Kannan and V. Lakshmikanthain (eds.), 2002, pp. 107–158.

    Google Scholar 

  41. H.-H. Kuo, A quarter century of white noise theory. Quantum Information IV, T. Hida and K. Saito (eds.), 2002, pp. 1–37.

    Google Scholar 

  42. E.W. Lytvynov, Multiple Wiener integrals and non-Gaussian white noises: a Jacobi field approach. Methods Funct. Anal. Topology 1 (1995), no. 1, 61–85.

    MATH  MathSciNet  Google Scholar 

  43. E. Lytvynov, Polynomials of Meixner’s type in infinite dimensions — Jacobi fields and orthogonality measures. J. Funct. Anal. 200 (2003), 118–149.

    Article  MATH  MathSciNet  Google Scholar 

  44. E. Lytvynov, Orthogonal decompositions for Levy processes with an applications to the Gamma, Pascal, and Meixner processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 1, 73–102.

    Article  MATH  MathSciNet  Google Scholar 

  45. E. Lytvynov, Functional spaces and operators connected with some Levy noises. arXiv:math.PR/0608380 vl 15 Aug 2006.

    Google Scholar 

  46. E.W. Lytvynov and G.F. Us, Dual Appell systems in non-Gaussian white noise calculus. Methods Funct. Anal. Topology 2 (1996), no. 2, 70–85.

    MATH  MathSciNet  Google Scholar 

  47. J. Merxner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. London Math. Soc. 9 (1934), no. 1, 6–13.

    Article  Google Scholar 

  48. I. Rodionova, Analysis connected with generating functions of exponential type in one and infinite dimensions. Methods Funct. Anal. Topology 11 (2005), no. 3, 275–297.

    MATH  MathSciNet  Google Scholar 

  49. V.A. Tesko, A construction of generalized translation operators. Methods Funct. Anal. Topology 10 (2004), no. 4, 86–92.

    MATH  MathSciNet  Google Scholar 

  50. V.A. Tesko, On spaces that arise in the construction of infinite-dimensional analysis according to a biorthogonal scheme. Ukrainian Math. J. 56 (2004), no. 7, 1166–1181.

    Google Scholar 

  51. G.F. Us, Dual Appel systems in Poissonian analysis. Methods Funct. Anal. Topology 1 (1995), no. 1, 93–108.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka St., 01601, Kyiv, Ukraine

    Yu.M. Berezansky & V.A. Tesko

Authors
  1. Yu.M. Berezansky
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  2. V.A. Tesko
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Vadim M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    Israel Gohberg

  3. Institute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivska st., 01601, Kyiv-4, Ukraine

    Anatoly Kochubei , Yurij Berezansky , Myroslav Gorbachuk  & Valentyna Gorbachuk , ,  & 

  4. Department of Mathematical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Gennadiy Popov

  5. Institute of Analysis and Scientific Computing, Technical University of Vienna, Wiedner Hauptstrasse 8-10, 1040, Vienna, Austria

    Heinz Langer

Additional information

This paper is dedicated to M. G. Krein

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© 2009 Birkhäuser Verlag Basel/Switzerland

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Berezansky, Y., Tesko, V. (2009). An Approach to a Generalization of White Noise Analysis. In: Adamyan, V.M., et al. Modern Analysis and Applications. Operator Theory: Advances and Applications, vol 190. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9919-1_7

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