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Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 919–946 | Cite as

Polynomial inequalities and universal Taylor series

  • Augustin MouzeEmail author
  • Vincent Munnier
Article

Abstract

We derive several new properties concerning both universal Taylor series and Fekete universal series from classical polynomial inequalities. In particular, we study some density properties of their approximating subsequences. Moreover we exhibit summability methods which preserve or imply the universality of Taylor series in the complex plane. Likewise we show that the partial sums of the Taylor expansion around zero of a \(C^{\infty }\) function is universal if and only if the sequence of its Cesàro means satisfies the same universal approximation property.

Keywords

Universal series Frequently universal series Bernstein inequality Density 

Mathematics Subject Classification

30K05 47A16 32A40 40A05 41A10 

Notes

Acknowledgments

We are indebted to the anonymous referee for useful comments and suggestions which considerably improved the presentation of the paper.

References

  1. 1.
    Aron, R., Beauzamy, B., Enflo, P.: Polynomials in many variables: real vs complex norms. J. Approx. Theory 74, 181–198 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayart, F.: Boundary behavior and Cesàro means of universal Taylor series. Rev. Math. Complut. 19(1), 235–247 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bayart, F., Grivaux, S.: Hypercyclicité : le rôle du spectre ponctuel unimodulaire. C. R. Math. Acad. Sci. Paris 338(9), 703–708 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358, 5083–5117 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bayart, F., Grosse-Erdmann, K.-G., Nestoridis, V., Papadimitropoulos, C.: Abstract theory of universal series and applications. Proc. Lond. Math. Soc. 96, 417–463 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Binmore, K.G.: On Turán’s lemma. Bull. Lond. Math. Soc. 3, 313–317 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergod. Theory Dyn. Syst. 27(2), 383–404 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics, vol. 161. Springer, New York (1995)zbMATHGoogle Scholar
  9. 9.
    Charpentier, S., Menet, Q., Mouze, A.: Closed universal subspaces of spaces of infinitely differentiable functions. Ann. Inst. Fourier (Grenoble) 64(1), 297–325 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Charpentier, S., Mouze, A.: Universal Taylor series and summability. Rev. Math. Complut. 28, 153–167 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chui, C.K., Parnes, M.N.: Approximation by overconvergence of power series. J. Math. Anal. Appl. 36, 693–696 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Costakis, G.: Which maps preserve universal functions?, Mathematisches Forschungsinstitut Oberwolfach, Report No 6/2008 (2008)Google Scholar
  13. 13.
    Durand, A.: Quelques Aspects de la Théorie Analytique des Polynômes I, II. Lectures Notes in Mathematics, vol. 1415, pp 43–85. Springer, Berlin (1990)Google Scholar
  14. 14.
    Freedman, A.R., Sember, J.J.: Densities and summability. Pac. J. Math. 95, 293–305 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gehlen, W., Luh, W., Müller, J.: On the existence of O-universal functions. Complex Var. Theory Appl. 41(1), 81–90 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grosse Erdmann, K.-G.: Holomorphe monster und universelle Funktionen. Mitt. Math. Sem. Giersen 176, 1–84 (1987)MathSciNetGoogle Scholar
  17. 17.
    Grosse Erdmann, K.-G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. 36(3), 345–381 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grosse Erdmann, K.-G., Peris, A.: Linear chaos. Universitext. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  19. 19.
    Katsoprinakis, E.S.: Coincidence of some classes of universal functions. Rev. Math. Complut. 22(2), 427–445 (2009)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Korevaar, J.: Tauberian Theory, A Century of Developments. A Series of Comprehensive Studies in Mathematics, vol. 329. Springer, Berlin Heidelberg (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Klimek, M.: Pluripotential Theory. London Mathematical Society Monographs, vol. 6. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1991)zbMATHGoogle Scholar
  22. 22.
    Luh, W.: Approximation analytischer Funktionen durch uberkonvergente Potenzreihen und deren Matrix-Transformierten. Mitt. Math. Sem. Giessen 88, 1–56 (1970)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Melas, A., Nestoridis, V.: Universality of Taylor series as a generic property of holomorphic functions. Adv. Math. 157(2), 138–176 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mouze, A., Munnier, V.: On the frequent universality of the classical universal Taylor series in the complex plane. Glasg. Math. J. (2016, to appear)Google Scholar
  25. 25.
    Mouze, A., Nestoridis, V.: Universality and ultradifferentiable functions: Fekete’s theorem. Proc. Am. Math. Soc. 138, 3945–3955 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nazarov, F.L.: Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra i Analiz 5(4), 3–66 (1993)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Nestoridis, V.: Universal Taylor series. Ann. Inst. Fourier (Grenoble) 46(5), 1293–1306 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pál, G.: Zwei kleine Bemerkungen. Tokohu Math. J. 6, 42–43 (1914/15)Google Scholar
  29. 29.
    Papachristodoulos, C.: Upper and lower frequently universal series. Glasg. Math. J. 55(3), 615–627 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rajagopal, C.T.: Some limit theorems. Am. J. Math. 70, 157–166 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Schaeffer, A.C., Duffin, R.J.: On some inequalities of S. Bernstein and W. Markoff for derivatives of polynomials. Bull. Am. Math. Soc. 44(4), 289–297 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Seleznev, A.I.: On universal power series. Math. Sbornik N.S. 28, 453–460 (1951)MathSciNetGoogle Scholar
  33. 33.
    Turán, P.: Eine neue Methode in der Analysis und deren Anwendungen. Akadémiai Kiadó, Budapest (1953)Google Scholar
  34. 34.
    Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (1979)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Laboratoire Paul PainlevéUMR 8524, Cité ScientifiqueVilleneuve d’Ascq CedexFrance
  2. 2.École Centrale de LilleCité ScientifiqueVilleneuve d’Ascq CedexFrance

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