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Orthogonal Approach to the Construction of the Theory of Generalized Functions of Infinitely Many Variables and the Poisson Analysis of White Noise

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Abstract

We develop an orthogonal approach to the construction of the theory of generalized functions of infinitely many variables (without using Jacobi fields) and apply it to the construction and investigation of the Poisson analysis of white noise.

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REFERENCES

  1. Yu. M. Berezans’kyi and V. A. Tesko, “Spaces of test and generalized functions related to generalized translation operators,” Ukr. Mat. Zh., 55, No.12, 1587–1657 (2003).

    Google Scholar 

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

    Google Scholar 

  3. Y. Ito and I. Kubo, “Calculus on Gaussian and Poisson white noise,” Nagoya Math. J., 111, 41–84 (1988).

    Google Scholar 

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

    Google Scholar 

  5. Yu. G. Kondratiev, J. L. da Silva, L. Streit, and G. F. Us, “Analysis on Poisson and Gamma spaces, ” Infinite Dim. Anal. Quantum Probab. Related Topics, 1, No.1, 91–117 (1998).

    Article  Google Scholar 

  6. S. Albeverio, Yu. G. Kondratiev, and M. Rockner, “Analysis and geometry on configuration spaces,” J. Funct. Anal., 154, No.2, 444–500 (1998).

    Article  Google Scholar 

  7. N. A. Kachanovsky, “On biorthogonal approach to a construction of non-Gaussian analysis and application to the Poisson analysis on the configuration space, ” Meth. Funct. Anal. Top., 6, No.2, 13–21 (2000).

    Google Scholar 

  8. T. Kuna, Studies in Configuration Space Analysis and Applications, Rheinische Friedrich-Wilhelms-Universitat, Bonn (1999).

    Google Scholar 

  9. Yu. G. Kondratiev, T. Kuna, and M. J. Oliveira, “Analytic aspects of Poissonian white noise analysis, ” Meth. Funct. Anal. Top., 8, No.4, 15–48 (2002).

    Google Scholar 

  10. M. J. Oliveira, Configuration Space Analysis and Poissonian White Noise Analysis, PhD Thesis, University of Lisbon, Lisbon (2002).

    Google Scholar 

  11. Yu. M. Berezans’kyi, “Poisson infinite-dimensional analysis as an example of analysis related to operators of generalized translation,” Funkts. Anal. Prilozhen., 32, No.3, 65–70 (1998).

    Google Scholar 

  12. E. W. Lytvynov, “A note on test and generalized functionals of Poisson white noise,” Hiroshima Math. J., 28, No.3, 463–480 (1998).

    MathSciNet  Google Scholar 

  13. Yu. G. Kondratiev and E. W. Lytvynov, “Operators of Gamma white noise calculus,” Infinite Dim. Anal. Quantum Probab. Related Topics, 3, No.3, 303–335 (2000).

    Google Scholar 

  14. Yu. M. Berezans’kyi and Yu. G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, Vols. 1, 2, Kluwer, Dordrecht (1995).

    Google Scholar 

  15. Yu. M. Berezans’kyi, “On images of operators of second quantization,” Dokl. Akad. Nauk Ukr., No. 11, 20–24 (1993).

    Google Scholar 

  16. D. L. Finkelshtein, Yu. G. Kondratiev, A. Yu. Konstantinov, and M. Rockner, “Symmetric differential operators of the second order in Poisson spaces,” Meth. Funct. Anal. Top., 7, No.4, 489–509 (2001).

    Google Scholar 

  17. Yu. M. Berezans’kyi, Z. G. Sheftel, and G. F. Us, Functional Analysis, Vols. 1, 2, Birkhauser, Basel (1996).

    Google Scholar 

  18. Yu. G. Kondratiev, J. L. da Silva, and L. Streit, “Generalized Appell systems,” Meth. Funct. Anal. Top., 3, No.3, 28–61 (1997).

    Google Scholar 

  19. Yu. G. Kondratiev, L. Streit, W. Westerkamp, and J. Yan, “Generalized functions on infinite dimensional analysis,” Hiroshima Math. J., 28, No.2, 213–260 (1998).

    Google Scholar 

  20. A. V. Skorokhod, Integration in a Hilbert Space [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  21. V. A. Tesko, “Spaces Appearing in the Construction of Infinite-Dimensional Analysis According to the Biorthogonal Scheme,” Ukr. Mat. Zh., 56, No.7, 977–990 (2004).

    Google Scholar 

  22. Yu. M. Berezans’kyi and A. A. Kalyuzhnyi, Harmonic Analysis in Hypercomplex Systems, Kluwer, Dordrecht (1998).

    Google Scholar 

  23. Yu. M. Berezans’kyi and Yu. G. Kondratiev, “Biorthogonal systems in hypergroups: an extension of non-Gaussian analysis,” Meth. Funct. Anal. Top., 2, No.2, 1–50 (1996).

    Google Scholar 

  24. V. G. Maz’ya and T. O. Shaposhnikova, Multiplicators in Spaces of Differentiable Functions [in Russian], Leningrad University, Leningrad (1986).

    Google Scholar 

  25. Yu. M. Berezans’kyi, “Poisson measure as the spectral measure of Jacobi field,” Infinite Dim. Anal. Quantum Probab. Related Topics, 3, No.1, 121–139 (2000).

    Article  Google Scholar 

  26. A. Lenard, “Correlation functions and the uniqueness of the state in classical statistical mechanics, ” Commun. Math. Phys., No. 30, 35–40 (1973).

  27. Yu. G. Kondratiev and T. Kuna, “Harmonic analysis on configuration space I. General theory,” Infinite Dim. Anal. Quantum Probab. Related Topics, 5, No.2, 201–233 (2002).

    Article  Google Scholar 

  28. J. Mecke, “Stationare zufallige Ma×e auf lokalkompakten Abelshen Gruppen,” Z. Wahrscheinlichkeitstheor. Verw. Geb., No. 9, 36–58 (1967).

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  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    Yu. M. Berezans’kyi & V. A. Tesko

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  1. Yu. M. Berezans’kyi
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  2. V. A. Tesko
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1587–1615, December, 2004.

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Berezans’kyi, Y.M., Tesko, V.A. Orthogonal Approach to the Construction of the Theory of Generalized Functions of Infinitely Many Variables and the Poisson Analysis of White Noise. Ukr Math J 56, 1885–1914 (2004). https://doi.org/10.1007/s11253-005-0158-y

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  • DOI: https://doi.org/10.1007/s11253-005-0158-y

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