Skip to main content
SpringerLink
Log in

Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

Pseudodifferential equations of the formv(Dx)y=f (here,v is a function holomorphic at zero andD x is a pseudodifferential operator) are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to the construction of a generalized translation operatorT χy =χ(<y, Dχ>) on the spaces indicated and to the investigation of its properties. In particular, we prove the associativity, commutativity, and other properties ofT χy , which are analogs of the classical properties of a generalized translation operator.

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  3. G. F. Us, “Dual Appell systems in Poissonian analysis,”Meth. Funct. Anal. Topol.,1, No. 1, 93–108 (1995).

    MATH  MathSciNet  Google Scholar 

  4. Yu. G. Kondrat’ev, L. Streit, W. Westerkamp, and J. Yan,Generalized Functions in Infinite-Dimensional Analysis, IIAS Preprint No. 1995-002, Kyoto (1995).

  5. S. Albeverio, Yu. L. Daletskii, Yu. G. Kondrat’ev, and L. Streit, “Non-Gaussian infinite-dimensional analysis,”J. Funct. Anal.,138, No. 2, 311–350 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  6. Yu. M. Berezanskii and Yu. G. Kondrat’ev, ”Biorthogonal systems in hypergroups: an extension of non-Gaussian analysis,”Meth. Funct. Anal. Topol.,2, No. 2, 1–50 (1996).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  8. N. A. Kachanovskii, “Biorthogonal Appell-like systems in a Hilbert space,”Meth. Funct. Anal. Topol.,2, No. 3,4, 36–52 (1996).

    MathSciNet  Google Scholar 

  9. N. A. Kachanovskii, “Dual Appell system and Kondrat’ev spaces in analysis on Schwartz spaces,”Ukr. Mat. Zh., No. 4, 527–534 (1997).

    MathSciNet  Google Scholar 

  10. Yu. M. Berezanskii, “Infinite-dimensional analysis related to generalized translation operators,”Ukr. Mat. Zh.,49, No. 3, 364–409 (1997).

    Article  MathSciNet  Google Scholar 

  11. N. A. Kachanovskii and G. F. Us, “Biorthogonal Appell systems in analysis on dually nuclear spaces,”Funkts. Anal. Prilozh.,32, No. 1, 69–72 (1998).

    MathSciNet  Google Scholar 

  12. Yu. M. Berezanskii, “Construction of generalized translation operators from the system of Appell characters,”Trans. Amer. Math. Soc.,184, No. 2, 7–21 (1998).

    MathSciNet  Google Scholar 

  13. R. P. Boas and R. C. Buck,Polynomial Expansions of Analytic Functions, Springer, Berlin (1964).

    MATH  Google Scholar 

Download references

Authors
  1. N. A. Kachanovskii
    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. 51, No. 10, pp. 1334–1341, October, 1999.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kachanovskii, N.A. Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis. Ukr Math J 51, 1503–1511 (1999). https://doi.org/10.1007/BF02981683

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02981683

Keywords

Search

Navigation