Skip to main content
SpringerLink
Log in

Infinite-dimensional analysis related to generalized translation operators

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We give an extensive generalization of the white-noise analysis (in the Gaussian and non-Gaussian case) in which the role of translation operators is played by a fixed family of generalized translation operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. Hida, Brownian Motion, Springer-Verlag, New York-Heidelberg-Berlin (1980).

    MATH  Google Scholar 

  2. T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise. An Infinite Dimensional Calculus, Kluwer, Dordrecht-Boston-London (1993).

    MATH  Google Scholar 

  3. Yu. M. Berezansky, Self-Adjoint Operators in the Spaces of Functions of Infinitely Many Variables [in Russian], Naukova Dumka, Kiev (1978). English translation: AMS, Providence (1986).

    Google Scholar 

  4. Yu. M. Berezansky and Yu. G. Kondrat’ev, Spectral Methods in Infinite-Dimensional Analysis [in Russian], Naukova Dumka, Kiev (1988), English translation: Kluwer, Dordrecht-Boston-London (1995).

    Google Scholar 

  5. Y. Itô, “Generalized Poisson functionals,” Probab. Theory Related Fields, 77, 1–28 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  6. Y. Itô and I. Kubo, “Calculus on Gaussian and Poisson white noises,” Nagoya Math. J., 111, 41–84 (1988).

    MATH  MathSciNet  Google Scholar 

  7. Yu. M. Berezansky, “Spectral approach to white noise analysis,” in: Proceedings of Symposium “Dynamics of Complex and Irregular Systems,” Bielefeld, December, 16–20, 1991; Bielefeld Encounters in Mathematics and Physics VIII, World Scientific, Singapore-New Jersey-London-Hong Kong (1993), pp. 131–140.

    Google Scholar 

  8. Yu. M. Berezansky, V. O. Livinsky, and E. W. Lytvynov, “A generalization of Gaussian white noise analysis,” Methods Funct. Analysis Topology, 1, No. 1, 28–55 (1995).

    MATH  MathSciNet  Google Scholar 

  9. E. W. Lytvynov, “Multiple Wiener integrals and non-Gaussian white noise analysis: a Jacobi field approach,” Methods Funct. Analysis Topology, 1, No. 1, 61–85 (1995).

    MATH  MathSciNet  Google Scholar 

  10. S. Albeverio, Yu. L. Daletsky, Yu. G. Kondratiev, and L. Streit, Non-Gaussian Infinite-Dimensional Analysis, BiBoS-Preprint, Bielefeld, 1994; J. Funct. Anal., 138, No. 2, 311–350 (1996).

  11. Yu. G. Kondratiev, L. Streit, W. Westerkamp, and J.-A. Yan, Generalized Functions in Infinite-Dimensional Analysis, IIAS Preprint (1995).

  12. G. F. Us, “Dual Appel systems in Poissonian anlysis,” Methods Funct. Analysis Topology, 1, No. 1, 93–108 (1995).

    MATH  MathSciNet  Google Scholar 

  13. Yu. L. Daletskii, “Biorthogonal analog of Hermitian polynomials and inversion of Fourier transformations with respect to non-Gaussian measure,” Funkts. Anal. Prilozh., 25, No. 2, 68–70 (1991).

    MathSciNet  Google Scholar 

  14. Yu. M. Berezansky and Yu. G. Kondrat’ev, “Non-Gaussian analysis and hypergroups,” Funkts. Anal. Prilozh., 29, No. 3, 51–55 (1995).

    Google Scholar 

  15. Yu. M. Berezansky, “A connection between the theory of hypergroups and white noise analysis,” Repts. Math. Phys., 36, No. 2/3, 215–234 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  16. Yu. M. Berezansky, “A generalization of white noise analysis by means of the theory of hypergroups,” Repts. Math. Phys., 38, No. 3, 289–300 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  17. Yu. M. Berezansky and Yu. G. Kondratiev, “Biorthogonal systems in hypergroups: an extension of non-Gaussian analysis,” Methods Funct. Analysis Topology, 2, No. 2, 1–50 (1996).

    MATH  MathSciNet  Google Scholar 

  18. Yu. M. Berezansky and A. A. Kalyuzhnyi, Harmonic Analysis in Hypercomplex Systems [in Russian], Naukova Dumka, Kiev (1992). English translation: Kluwer, Dordrecht-Boston-London (1997).

    Google Scholar 

  19. W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin-New York (1995).

    MATH  Google Scholar 

  20. B. M. Levitan, Theory of Generalized Translation Operators [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  21. Yu. M. Berezansky, “Infinite-dimensional non-Gaussian analysis and generalized translation operators,” Funkts. Anal. Prilozh., 30, No. 4, 61–65 (1996).

    MathSciNet  Google Scholar 

  22. Yu. M. Berezansky, V. O. Livinsky and E. V. Lytvynov, “Spectral approach to the white noise analysis,” Ukr. Mat. Zh., 46, No. 3, 177–197 (1994).

    Google Scholar 

  23. L. Nachbin, Topology on Spaces of Holomorphic Mappings, Springer-Verlag, Berlin-Heidelberg-New York (1969).

    MATH  Google Scholar 

Download references

Authors
  1. Yu. M. Berezansky
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 364–409, March, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berezansky, Y.M. Infinite-dimensional analysis related to generalized translation operators. Ukr Math J 49, 403–450 (1997). https://doi.org/10.1007/BF02487241

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02487241

Keywords

Search

Navigation