Skip to main content
SpringerLink
Log in

The problem of differentiating an asymptotic expansion in real powers. Part I: Unsatisfactory or partial results by classical approaches

О првблеме дифференцирования асимптотического разложения с вещественными покажателями. Часть I: Нердовлетворительные или частичные режулвтаты при классическом подходе

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

In two papers, the problem of formal differentiation of an asymptotic expansion in the real domain of type

$$ f(x) - a_1 x^{\alpha _1 } + \cdots + a_n x^{\alpha _n } + o(x^{\alpha _n } ),x \to + \infty , $$

is amply studied. In Part I, we show that the classical viewpoints and techniques concerning formal differentiation of an asymptotic relation

$$ f(x) - ax^\alpha + o(x^\alpha ),x \to + \infty , $$

give either unsatisfactory or partial results when applied to an asymptotic expansion with at least two meaningful terms. Simple examples show that some of these results are the best possible in the classical context. Hence a change of viewpoint is necessary to arrive at useful results.

Режюме

В лврх статьях подобно ижучается проблема формального дифференцированил асимптотического раэлозения вида

$$ f(x) - a_1 x^{\alpha _1 } + \cdots + a_n x^{\alpha _n } + o(x^{\alpha _n } ),x \to + \infty , $$

в веЩественнои области.

В части I показано, что классическая точка зрения и техника формального дифференцирования одночленного асимптотического соотноШения

$$ f(x) - ax^\alpha + o(x^\alpha ),x \to + \infty , $$

приводят либо к неудовлетворительным, либо только частичным результатам, если их применять к асимптотическому разложению с парои значимых членов. Простые нримеры иллюстрируит, что некоторые из этих результатов — наилучщие в классическом контексте. Поэтому чтобы получить полезные реультаты, нржно изменить точку зрения.

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 for parts I–II

  1. L. Berg, Differentiation asymptotischer Beziehungen, Math. Nachr., 29(1965), 123–129.

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Berg, Asymptotische Darstellungen und Entwicklungen, VEB Deutscher Verlag der Wissenshaften (Berlin, 1968).

    MATH  Google Scholar 

  3. R. P. Boas, JR., Asymptotic relations for derivatives, Duke Math. J., 3(1937), 637–646.

    Article  MathSciNet  Google Scholar 

  4. R. P. Boas, Jr., A Tauberian theorem connected with the problem of three bodies, Amer. Math. J., 61(1939), 161–164.

    Article  MathSciNet  Google Scholar 

  5. N. G. de Bruijn, Asymptotic methods in analysis, North-Holland Publishing Company and Wolters-Noordhoff Publishing (1970); corrected reprint of the third edition: Dover Publications Inc.(New York, 1981).

  6. E. Escangon, Nouvelles recherches sur les fonctions quasi-periodiques, Annales de l’Observatoire de Bordeaux, XVI(1917), 51–226.

    Google Scholar 

  7. A. Granata, Singular Cauchy problems and asymptotic behaviour for a class of n-th order differential equations, Funkcial. Ekvac., 20(1977), 193–212.

    MATH  MathSciNet  Google Scholar 

  8. A. Granata, Canonical factorizations of disconjugate differential operators, SIAM J. Math. Anal., 11(1980), 160–172.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Granata, Canonical factorizations of disconjugate differential operators. Part II, SIAM J. Math. Anal., 19(1988), 1162–1173.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Granata, Polynomial asymptotic expansions in the real domain: the geometric, the factorizational, and the stabilization approaches, Analysis Mathematica., 33(2007), 161–198.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. H. Hardy, Orders of infinity, Cambridge University Press (Cambridge, 1924).

    MATH  Google Scholar 

  12. G. H. Hardy, Divergent series, Oxford University Press (Oxford, 1949); (Reprinted in 1973).

    MATH  Google Scholar 

  13. G. H. Hardy and J. E. Littlewood, Contributions to the arithmetic theory of series, Proc. London Math. Soc. (2), 11(1913), 411–478.

    Article  Google Scholar 

  14. G. H. Hardy and J. E. Littlewood, Tauberian theorems concerning power series and Dirichlet’s series whose coefficients are positive, Proc. London Math. Soc. (2), 13(1914), 174–191.

    Article  Google Scholar 

  15. E. Hewitt and K. Stromberg, Real and abstract analysis, Springer (Berlin-Heidelberg-New York, 1969.)

    MATH  Google Scholar 

  16. J. Karamata, Über einen Tauberschen Satz im Dreikörperproblem, Amer. J. Math., 61(1939), 769–770.

    Article  MathSciNet  Google Scholar 

  17. S. Karlin and W. Studden, Tchebycheff systems: with applications in analysis and statistics, Interscience (New York, 1966).

    MATH  Google Scholar 

  18. E. Landau, Beiträge zur analytischen Zahlentheorie, Rend. Circ. Mat. Palermo, 26(1908), 169–302.

    Article  MATH  Google Scholar 

  19. E. Landau, Darstellung und Begründung einiger neueren Ergebnisse der Funktionentheorie, 2nd ed., Springer (Berlin, 1929); reprinted in: E. Weyl, E. Landau, and B. Reimann, Das Kontinuum und andere Monographien, Chelsea Publ. Co. (New York, 1960).

    Google Scholar 

  20. E. Landau, Über einen Satz von Herrn Esclangon, Math. Ann., 102(1929–30), 177–188.

    Article  MATH  Google Scholar 

  21. S. Leader, The Kurzweil-Henstock integral and its differentials, Marcel Dekker Inc. (New York-Basel, 2001).

    MATH  Google Scholar 

  22. A. Yu. Levin, Non-oscillation of solutions of the equation x(n) + p1(t)x(n−1) + ... + pn(t)x = 0, Uspekhi Mat. Nauk, 24(1969), no. 2 (146), 43–96; Russian Math. Surveys, 24(1969), no. 2, 43–99.

    Google Scholar 

  23. J. E. Littlewood, The converse of Abel’s theorem on power series, Proc. London Math. Soc. (2), 9(1910–1911), 434–448. See also “Comments on the preceding paper”, p. 772 in Collected papers by J. E. Littlewood, vol. I. Clarendon Press, Oxford University press (New York, 1982).

    Article  MathSciNet  Google Scholar 

  24. J. E. Littlewood, A theorem about successive derivatives of a function and some Tauberian theorems, J. London Math. Soc., 42(1967), 169–179.

    Article  MATH  MathSciNet  Google Scholar 

  25. D. G. Saari, Expanding gravitational systems, Trans. Amer. Math. Soc., 156(1971), 219–240.

    Article  MATH  MathSciNet  Google Scholar 

  26. D. G. Saari, An elementary Tauberian theorem for absolutely continuous functions and for series, SIAM J. Math. Anal., 5(1974), 649–662.

    Article  MATH  MathSciNet  Google Scholar 

  27. W. F. Trench, Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc., 189(1974), 139–327.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università della Calabria, 87036, Rende (Cosenza), Italy

    Antonio Granata

Authors
  1. Antonio Granata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Granata.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Granata, A. The problem of differentiating an asymptotic expansion in real powers. Part I: Unsatisfactory or partial results by classical approaches. Anal Math 36, 85–112 (2010). https://doi.org/10.1007/s10476-010-0201-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-010-0201-6

Keywords

Search

Navigation