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A Version of Lagrange's Theorem for Some Classes of Functions of Many Variables

Year 2014, Volume: 5 , 1 - 8, 06.03.2015

Abstract

The famous mean motion problem which goes back to Lagrange as
follows: to prove that any exponential polynomial with exponents
on the imaginary axis has an average speed for the amplitude,
whenever the variable moves along a horizontal line. It was
completely proved by B.\,Jessen and H.\,Tornehave in Acta Math.77,
1945. Actually, this result is a consequence of almost periodicity
in Weyl's sense of amplitude increments over segments of the
length 1. Here we consider the problem for some classes of almost
periodic functions of several variables.

References

  • Besicovitch A.C., Almost periodic functions, Cambridge university press, 1932.
  • Bohr H., Kleinere Beiltrage zur Theorie der fastperiodischen Funktionen, I, Acta math., 45 (1924), 29–127.
  • Bernstein F., ¨ Uber eine Anwendung der Mengenlehre auf ein aus der Theorie der s¨ akularen St¨ orungen herr¨ uhrendes Problem, Math. A00nn., 77 (1912), 417–439.
  • Favorov S.Yu., Holomorphic almost periodic functions in tube domains and their amoebas, Computational Methods and Function Theory, 1(2) (2001), 403–415.
  • Favorov S.Yu., Lagrange’s Problem on Mean Motion, Algebra and Analyse, 20(2) (2008), 218–225 (Russian).
  • Favorov S.Yu, Girya N.P., A multidimensional version of Levin’s Secular Constant Theorem and its Applications, Journal of Mathematical Physics, Analysis, Geometry, 3(3) (2007), 365–377.
  • Gelfond O.A, Roots of systems of almost periodic polynomials, Preprint FIAN SSSR, no. 200, 1978 (Russian).
  • Herve M., Several Complex Variables. Local Theory, Oxford University Press, Bombay, 1963.
  • Jessen B., ¨ Uber die Nullstellen einer analitischen fastperiodischen Funktions, Eine Verallagemeinerung der Jensenschen Formel, Math. Ann., 108 (1933), 485–516.
  • Jessen B., Tornehave H., Mean motions and zeros of almost periodic functions, Acta Math., 77 (1945), 137–279.
  • Kazarnovskii B.Ja., Zeros of exponential sums, Dokl. AN SSSR, 257(4) (1981), 804–808 (Russian).
  • Kazarnovskii B.Ja., Newton polyhedron and roots of exponential sums, Funk. Analis i pril., 18(4) (1984), 40–49 (Russian).
  • Lagrange J.L., Th´ eorie des variations s´ equlaires des ´ el´ ements des plan` etes I, II, Nouveaux M´ emorires de l’Acad´ emie de Berlin, (1781-1782), 123–344.
  • Forsberg M.,Passare M., Tsikh A., Laurent determinants and arrangement of hyperplane amoebas, Adv. Math., 151(1) (2000), 45–70.
  • Ronkin L.I., Jessen’s theorem for holomorphic almost periodic functions in tube domains, Sib.Math.Zhurn, 28(3) (1987), 199–204 (Russian).
  • Ronkin L.I., On a certain class of holomorphic almost periodic functions, Sib.Math.Zhurn, 33 (1992), 135–141 (Russian).
  • Ronkin L.I., Introduction to the theory of entire functions of several variables, Translations of Math. Monographs, V.44, AMS, Providence, R.1, 1974.
  • Udodova O.I., Holomorphic almost periodic functions in various metrics, Vestnik of KhNU, Ser. ”Mathematics, Applied Mathematics, and Mechanics”, 52(582) (2003), 90–107 (Russian).
  • Weyl H., ¨ Uber die Gleichverteilung von Zahlen mod Eins, Math.Ann., 77 (1916), 313–352.
  • Weyl H., Mean Motion I, Amer.J.Math., 60 (1938), 889–896. Sergey Yu. Favorov, The Karazin Kharkiv National University, Svoboda squ., 4, 61022, Kharkiv, Ukraine, e-mail : sfavorov@gmail.com Natalya P. Girya, The Karazin Kharkiv National University, Svoboda squ., 4, 61022, Kharkiv, Ukraine, e-mail : n girya@mail.ru
Year 2014, Volume: 5 , 1 - 8, 06.03.2015

Abstract

References

  • Besicovitch A.C., Almost periodic functions, Cambridge university press, 1932.
  • Bohr H., Kleinere Beiltrage zur Theorie der fastperiodischen Funktionen, I, Acta math., 45 (1924), 29–127.
  • Bernstein F., ¨ Uber eine Anwendung der Mengenlehre auf ein aus der Theorie der s¨ akularen St¨ orungen herr¨ uhrendes Problem, Math. A00nn., 77 (1912), 417–439.
  • Favorov S.Yu., Holomorphic almost periodic functions in tube domains and their amoebas, Computational Methods and Function Theory, 1(2) (2001), 403–415.
  • Favorov S.Yu., Lagrange’s Problem on Mean Motion, Algebra and Analyse, 20(2) (2008), 218–225 (Russian).
  • Favorov S.Yu, Girya N.P., A multidimensional version of Levin’s Secular Constant Theorem and its Applications, Journal of Mathematical Physics, Analysis, Geometry, 3(3) (2007), 365–377.
  • Gelfond O.A, Roots of systems of almost periodic polynomials, Preprint FIAN SSSR, no. 200, 1978 (Russian).
  • Herve M., Several Complex Variables. Local Theory, Oxford University Press, Bombay, 1963.
  • Jessen B., ¨ Uber die Nullstellen einer analitischen fastperiodischen Funktions, Eine Verallagemeinerung der Jensenschen Formel, Math. Ann., 108 (1933), 485–516.
  • Jessen B., Tornehave H., Mean motions and zeros of almost periodic functions, Acta Math., 77 (1945), 137–279.
  • Kazarnovskii B.Ja., Zeros of exponential sums, Dokl. AN SSSR, 257(4) (1981), 804–808 (Russian).
  • Kazarnovskii B.Ja., Newton polyhedron and roots of exponential sums, Funk. Analis i pril., 18(4) (1984), 40–49 (Russian).
  • Lagrange J.L., Th´ eorie des variations s´ equlaires des ´ el´ ements des plan` etes I, II, Nouveaux M´ emorires de l’Acad´ emie de Berlin, (1781-1782), 123–344.
  • Forsberg M.,Passare M., Tsikh A., Laurent determinants and arrangement of hyperplane amoebas, Adv. Math., 151(1) (2000), 45–70.
  • Ronkin L.I., Jessen’s theorem for holomorphic almost periodic functions in tube domains, Sib.Math.Zhurn, 28(3) (1987), 199–204 (Russian).
  • Ronkin L.I., On a certain class of holomorphic almost periodic functions, Sib.Math.Zhurn, 33 (1992), 135–141 (Russian).
  • Ronkin L.I., Introduction to the theory of entire functions of several variables, Translations of Math. Monographs, V.44, AMS, Providence, R.1, 1974.
  • Udodova O.I., Holomorphic almost periodic functions in various metrics, Vestnik of KhNU, Ser. ”Mathematics, Applied Mathematics, and Mechanics”, 52(582) (2003), 90–107 (Russian).
  • Weyl H., ¨ Uber die Gleichverteilung von Zahlen mod Eins, Math.Ann., 77 (1916), 313–352.
  • Weyl H., Mean Motion I, Amer.J.Math., 60 (1938), 889–896. Sergey Yu. Favorov, The Karazin Kharkiv National University, Svoboda squ., 4, 61022, Kharkiv, Ukraine, e-mail : sfavorov@gmail.com Natalya P. Girya, The Karazin Kharkiv National University, Svoboda squ., 4, 61022, Kharkiv, Ukraine, e-mail : n girya@mail.ru
There are 20 citations in total.

Details

Primary Language English
Journal Section Mathematics
Authors

Sergey Favorov

Natalya Girya This is me

Publication Date March 6, 2015
Published in Issue Year 2014 Volume: 5

Cite

APA Favorov, S., & Girya, N. (2015). A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables. İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy, 5, 1-8.
AMA Favorov S, Girya N. A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables. İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy. March 2015;5:1-8.
Chicago Favorov, Sergey, and Natalya Girya. “A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables”. İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy 5, March (March 2015): 1-8.
EndNote Favorov S, Girya N (March 1, 2015) A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables. İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy 5 1–8.
IEEE S. Favorov and N. Girya, “A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables”, İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy, vol. 5, pp. 1–8, 2015.
ISNAD Favorov, Sergey - Girya, Natalya. “A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables”. İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy 5 (March 2015), 1-8.
JAMA Favorov S, Girya N. A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables. İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy. 2015;5:1–8.
MLA Favorov, Sergey and Natalya Girya. “A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables”. İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy, vol. 5, 2015, pp. 1-8.
Vancouver Favorov S, Girya N. A Version of Lagrange’s Theorem for Some Classes of Functions of Many Variables. İstanbul University Science Faculty the Journal of Mathematics Physics and Astronomy. 2015;5:1-8.