Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 1185–1197 | Cite as

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

  • Hans-Peter Beise
  • Jürgen MüllerEmail author


It is known that, generically, Taylor series of functions holomorphic in the unit disc turn out to be universal series outside of the unit disc and in particular on the unit circle. Due to classical and recent results on the boundary behaviour of Taylor series, for functions in Hardy spaces and Bergman spaces the situation is drastically different. In this paper it is shown that in many respects these results are sharp in the sense that universality generically appears on maximal exceptional sets. As a main tool it is proved that the Taylor (backward) shift on certain Bergman spaces is mixing.


Backward shift Mixing operator Universality 

Mathematics Subject Classification

30B30 30K05 47A16 


  1. 1.
    Anderson, J.M., Clunie, J., Pommerenke, C.: On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12–37 (1974)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bayart, F., Grosse-Erdmann, K.-G., Nestoridis, V., Papadimitropoulos, C.: Abstract theory of universal series and applications. Proc. Lond. Math. Soc. 96(3), 417–463 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Beise, H.-P., Meyrath, T., Müller, J.: Mixing Taylor shifts and universal Taylor series. Bull. Lond. Math. Soc. 47, 136–142 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bers, L.: An approximation theorem. J. Anal. Math. 14, 1–4 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cima, J.A., Ross, W.T.: The backward shift on the Hardy space. American Mathematical Society, Providence (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Conway, J.B.: Functions of a Complex Variable II. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  8. 8.
    Duren, P.: Theory of \(H^p\) spaces. Dover, Mineola (2000)Google Scholar
  9. 9.
    Duren, P., Schuster, A.: Bergman Spaces, Mathematical surveys and monographs, no. 100. American Mathematical Society, Providence (2000)Google Scholar
  10. 10.
    Garnett, J.B.: Bounded Analytic Functions. Springer, New York (2007)zbMATHGoogle Scholar
  11. 11.
    Gardiner, S., Manolaki, M.: A convergence theorem for harmonic measures with applications to Taylor series. Proc. Am. Math. Soc. 144, 1109–1117 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gethner, R.M., Shapiro, J.H.: Universal vectors for operators on spaces of holomorphic functions. Proc. Am. Math. Soc. 100, 281–288 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grosse-Erdmann, K.G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. 36, 345–381 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grosse-Erdmann, K.G., Peris, M.A.: Linear Chaos. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    Havin, P.: Analytic representation of linear functionals in spaces of harmonic and analytic functions continuous in a closed region. Dokl. Akad. Nauk SSSR 151, 505–508 (1963), English transl. in Soviet Math. Dokl. 4 (1963)Google Scholar
  16. 16.
    Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  17. 17.
    Herzog, G., Kunstmann, P.: Universally divergent Fourier series via Landau’s extremal functions. Comment. Math. Univ. Carolin. 56, 159–168 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hruščev, S.V.: The problem of simultaneous approximation and removal of singularities of Cauchy type integrals. (Russian) Spectral theory of functions and operators. Trudy Mat. Inst. Steklov. 130, 124–195 (1978), English transl. in Proceedings of the Steklov Institute of Mathematics 4 (1979)Google Scholar
  19. 19.
    Hruščev, S.V., Peller, V.: Hankel operators, best approximation, and stationary Gaussian processes. Russ. Math. Surv. 67, 61–144 (1982)zbMATHGoogle Scholar
  20. 20.
    Kahane, J.P.: The Baire category theorem and trigonometric series. J. Anal. Math. 80, 143–182 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Katsoprinakis, E., Nestoridis, V., Papachristodoulos, C.: Universality and Cesàro summability. Comput. Methods Funct. Theory 12, 419–448 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Katznelson, Y.: An Introduction to Harmonic Analysis, 2nd edn. Dover, New York (1976)zbMATHGoogle Scholar
  23. 23.
    Müller, J.: Continuous functions with universally divergent Fourier series on small subsets of the circle. C. R. Math. Acad. Sci. Paris 348, 1155–1158 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Offord, A.C.: On the summability of power series. Proc. Lond. Math. Soc. S233, 467–480Google Scholar
  25. 25.
    Papachristodoulos, C., Papadimitrakis, M.: On universality and convergence of the Fourier series of functions in the disc algebra, to appear in J. Anal. Math. arXiv:1503.03426v2
  26. 26.
    Remmert, R.: Classical Topics in Complex Function Theory. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  27. 27.
    Ross, W.T.: The backward shift on \(H^p\). Oper. Theory Adv. Appl. 158, 191–211 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shkarin, S.: Pointwise universal trigonometric series. J. Math. Anal. Appl. 360, 754–758 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zygmund, A.: Trigonometric Series, vol. I. Cambridge University Press, Cambridge (1977)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.FB IV, MathematicsUniversity of TrierTrierGermany

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