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Taylor Domination, Difference Equations, and Bautin Ideals

  • Dmitry Batenkov
  • Yosef Yomdin
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 180)

Abstract

We compare three approaches to studying the behavior of an analytic function \(f(z)=\sum _{k=0}^\infty a_kz^k\) from its Taylor coefficients. The first is “Taylor domination” property for f(z) in the complex disk \(D_R\), which is an inequality of the form
$$ |a_{k}|R^{k}\le C\ \max _{i=0,\dots ,N}\ |a_{i}|R^{i}, \ k \ge N+1. $$
The second approach is based on a possibility to generate \(a_k\) via recurrence relations. Specifically, we consider linear non-stationary recurrences of the form
$$ a_{k}=\sum _{j=1}^{d}c_{j}(k)\cdot a_{k-j}, \ k=d,d+1,\dots , $$
with uniformly bounded coefficients. In the third approach we assume that \(a_k=a_k(\lambda )\) are polynomials in a finite-dimensional parameter \(\lambda \in {\mathbb C}^n.\) We study “Bautin ideals” \(I_k\) generated by \(a_{1}(\lambda ),\ldots ,a_{k}(\lambda )\) in the ring \({\mathbb C}[\lambda ]\) of polynomials in \(\lambda \). These three approaches turn out to be closely related. We present some results and questions in this direction.

Keywords

Recurrence relations Bautin ideals Domination of initial Taylor coefficients 

References

  1. 1.
    Batenkov, D.: Moment inversion problem for piecewise D-finite functions. Inverse Problems 25(10), 105001 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Batenkov, D., Binyamini, G.: Moment vanishing of piecewise solutions of linear ODE’s. arXiv:1302.0991. (Submitted to this volume)
  3. 3.
    Batenkov, D., Sarig, N., Yomdin, Y.: Accuracy of Algebraic Fourier reconstruction for shifts of several signals. Sampl. Theory Signal Image Process. 13, 151–173 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Batenkov, D., Yomdin, Y.: Taylor Domination, Turán lemma, and Poincaré-Perron Sequences. Contemp. Math. 659, 1–15 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bautin, N.: Du nombre de cycles limites naissant en cas de variation des coefficients d’un etat d’equilibre du type foyer ou centre. C. R. (Doklady) Acad. Sci. URSS (N. S.) 24, 669–672 (1939)Google Scholar
  6. 6.
    Bautin, N.: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Am. Math. Soc. Transl. 100, 19 (1954)Google Scholar
  7. 7.
    Bieberbach, L.: Analytische Fortsetzung. Springer, Berlin (1955)CrossRefzbMATHGoogle Scholar
  8. 8.
    Biernacki, M.: Sur les fonctions multivalentes d’ordre p. CR Acad. Sci. Paris 203, 449–451 (1936)zbMATHGoogle Scholar
  9. 9.
    Blinov, M., Briskin, M., Yomdin, Y.: Local center conditions for Abel equation and cyclicity of its zero solution. Complex analysis and dynamical systems II, Contemp. Math. Amer. Math. Soc. Providence, RI, 382, 65–82 (2005)Google Scholar
  10. 10.
    Bodine, S., Lutz, D.A.: Asymptotic solutions and error estimates for linear systems of difference and differential equations. J. Math. Anal. Appl. 290(1), 343–362 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Borcea, J., Friedland, S., Shapiro, B.: Parametric Poincaré-Perron theorem with applications. J. d’Analyse Mathématique 113(1), 197–225 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Briskin, M., Roytvarf, N., Yomdin, Y.: Center conditions at infinity for Abel differential equations. Ann. Math. 172(1), 437–483 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Briskin, M., Yomdin, Y.: Algebraic families of analytic functions I. J. Differ. Equ. 136(2), 248–267 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Coppel, W.: Dichotomies and stability theory. In: Proceedings of the Symposium on Differential Equations and Dynamical Systems, pp. 160–162. Springer, Heidelberg (1971)Google Scholar
  15. 15.
    Francoise, J.-P., Yomdin, Y.: Bernstein inequality and applications to analytic geometry and differential equations. J. Funct. Anal. 146(1), 185–205 (1997)Google Scholar
  16. 16.
    Friedland, O., Yomdin, Y.: An observation on Turán-Nazarov inequality. Studia Math. 218(1), 27–39 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Friedland, O., Yomdin, Y.: \((s, p)\)-valent functions, to appear in GAFA seminar notesGoogle Scholar
  18. 18.
    Hayman, W.K.: Multivalent functions, vol. 110. Cambridge University Press, Cmabridge (1994)Google Scholar
  19. 19.
    Kloeden, P., Potzsche, C.: Non-autonomous difference equations and discrete dynamical systems. J. Differ. Equ. Appl. 17(2), 129–130 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mitrinovic, D.S., Pecaric, J., Fink, A.M.: Classical and New Inequalities in Analysis, Series: Mathematics and its Applications, vol. 61, XVIII, p. 740 (1993)Google Scholar
  21. 21.
    Nazarov, F.L.: Local estimates of exponential polynomials and their applications to inequalities of uncertainty principle type. St Petersb. Math. J. 5(4), 663–718 (1994)MathSciNetGoogle Scholar
  22. 22.
    Perron, O.: Über summengleichungen und Poincarésche differenzengleichungen. Mathematische Annalen 84(1), 1–15 (1921)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pituk, M.: More on Poincaré’s and Perron’s Theorems for Difference Equations. J. Differ. Equ. Appl. 8(3), 201–216 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Poincare, H.: Sur les équations linéaires aux différentielles ordinaires et aux différences finies. Am. J. Math. 7(3), 203–258 (1885)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pötzsche, C.: Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics. Springer, Berlin (2010)Google Scholar
  26. 26.
    Roytwarf, N., Yomdin, Y.: Bernstein classes. Annales de l’institut Fourier 47, 825–858 (1997)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Turán, P.: Eine neue Methode in der Analysis und deren Anwendungen. Akadémiai Kiadó, 1953Google Scholar
  28. 28.
    Turán, P., Halász, G., Pintz, J.: On a new method of analysis and its applications. Wiley-Interscience, New York (1984)Google Scholar
  29. 29.
    Yomdin, Y.: Nonautonomous linearization. Dynamical Systems (College Park, MD). Lecture Notes in Mathematics, vol. 1342, pp. 718–726. Springer, Berlin (1988)Google Scholar
  30. 30.
    Yomdin, Y.: Global finiteness properties of analytic families and algebra of their Taylor coefficients. The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun., vol. 24. Amer. Math. Soc., Providence, RI, pp. 527–555 (1999)Google Scholar
  31. 31.
    Yomdin, Y.: Singularities in algebraic data acquisition. Real and Complex Singularities. London Math. Soc. Lecture Notes Series, vol. 380, pp. 378–396, Cambridge Univeristy Press, Cambridge (2010)Google Scholar
  32. 32.
    Yomdin, Y.: Bautin ideals and Taylor domination. Publ. Mat. 58, 529–541 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of MathematicsWeizmann Institute of ScienceRehovotIsrael

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