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On the integral of a function along the trajectories of a nilpotent flow

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Abstract

We establish conditions under which the integral of a function along a nilpotent flow on the Heisenberg-Iwasawa manifold increases not faster than\(|t|^{{1 \mathord{\left/ {\vphantom {1 {2 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {2 + \varepsilon }}} ,0< \varepsilon<< 1\) and indicate cases where this integral can be represented as a superposition of a function defined on a nilmanifold and a nilpotent flow.

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References

  1. L. Auslander, L. Green, and F. Hahn,Flows on Homogeneous Spaces, Annals of Mathematics Studies, Vol. 53, Princeton University Press, Princeton, New Jersey (1963).

    Google Scholar 

  2. M. I. Kadets, “Integration of almost periodic functions with values in Banach spaces,”Funkts. Anal. Prilozhen.,3, Issue 3, 71–74 (1969).

    Google Scholar 

  3. A. M. Samoilenko,Elements of the Mathematical Theory of Multifrequency Oscillations. Invariant Tori [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  4. V. V. Kozlov, “On integrals of quasiperiodic functions,”Vestn. Most Univ., Ser. Mat. Mekh., No. 1, 31–40 (1978).

    Google Scholar 

  5. V. V. Kozlov,Methods of Qualitative Analysis in the Dynamics of Solid Bodies [in Russian], Moscow University, Moscow (1980).

    Google Scholar 

  6. N. G. Moshchevitin, “Behavior of the integral of a conditionally periodic function,”Mat. Zametki,50, No. 3, 97–106 (1991).

    Google Scholar 

  7. I. M. Vinogradov, “Method of trigonometric sums in number theory,” in:Selected Works [in Russian], Izd. Akad. Nauk SSSR, Moscow (1952), pp. 237–331.

    Google Scholar 

  8. S. Lang,Introduction to Diophantine Approximations, Addison-Wesley, Reading, Massachusetts (1966).

    Google Scholar 

  9. V. G. Sprindzhuk,Metric Theory of Diophantine Approximations [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  10. V. K. Dzyadyk,Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  11. N. P. Korneichuk,Extremal Problems in the Theory of Approximation [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  12. H. Rüssmann, “Notes on sums containing small divisors,”Commun. Pure Appl. Math.,24, No. 6, 755–758 (1976).

    Google Scholar 

  13. V. I. Arnol'd, “Proof of Kolmogorov's theorem on preservation of conditionally periodic motions under small perturbations of the Hamilton function,”Usp. Mat. Nauk,18, Issue 5, 13–40 (1969).

    Google Scholar 

  14. N. N. Bogolyubov, Yu. A. Mitropolskii, and A. M. Samoilenko,Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    A. M. Samoilenko (Academician of the Ukrainian Academy of Sciences)

  2. Kiev University, Kiev

    I. O. Parasyuk

Authors
  1. A. M. Samoilenko
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  2. I. O. Parasyuk
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 837–847, June, 1995.

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Samoilenko, A.M., Parasyuk, I.O. On the integral of a function along the trajectories of a nilpotent flow. Ukr Math J 47, 963–975 (1995). https://doi.org/10.1007/BF01058785

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  • DOI: https://doi.org/10.1007/BF01058785

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