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Families of strongly annular functions: linear structure

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Abstract

A function f holomorphic in the unit disk \(\mathbb{D}\) is called strongly annular if there exists a sequence of concentric circles in \(\mathbb{D}\) expanding out to the unit circle such that f goes to infinity as |z| goes to 1 through these circles. The residuality of the family of strongly annular functions in the space of holomorphic functions on \(\mathbb{D}\) is well known, and it is extended here to certain classes of functions. This important topological property is enriched in this paper by studying algebraic-topological properties of the mentioned family, in the modern setting of lineability. Namely, we prove that although this family is clearly nonlinear, it contains, except for the zero function, large vector subspaces as well as infinitely generated algebras. Similar results are obtained for strongly annular functions on the whole complex plane and for weighted Bergman spaces.

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References

  1. Aron, R.M., Seoane-Sepúlveda, J.B.: Algebrability of the set of everywhere surjective functions on ℂ. Bull. Belg. Math. Soc. Simon Stevin 13, 1–7 (2006)

    MathSciNet  Google Scholar 

  2. Aron, R., García, D., Maestre, M.: Linearity in non-linear problems. Rev. R. Acad. Cien. Ser. A Mat. 95, 7–12 (2001)

    MATH  Google Scholar 

  3. Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on ℝ. Proc. Am. Math. Soc. 133, 795–803 (2005)

    Article  MATH  Google Scholar 

  4. Aron, R.M., Pérez-García, D., Seoane-Sepúlveda, J.B.: Algebrability of the set of non-convergent Fourier series. Stud. Math. 175, 83–90 (2006)

    Article  MATH  Google Scholar 

  5. Aron, R., García-Pacheco, F.J., Pérez-García, D., Seoane-Sepúlveda, J.B.: On dense-lineability of sets of functions on R. Topology 48, 149–156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barth, K., Bonar, D., Carroll, F.: Zeros of strongly annular functions. Math. Z. 144, 175–179 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bayart, F.: Topological and algebraic genericity of divergence and universality. Stud. Math. 167, 161–181 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Belna, C.L., Redett, D.A.: A residual class of holomorphic functions. Comput. Methods Funct. Theory 10, 207–213 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Bernal-González, L.: Dense-lineability in spaces of continuous functions. Proc. Am. Math. Soc. 136, 3163–3169 (2008)

    Article  MATH  Google Scholar 

  10. Bernal-González, L.: Algebraic genericity of strict order integrability. Stud. Math. 199, 279–293 (2010)

    Article  MATH  Google Scholar 

  11. Boas, R.P.: Entire Functions. Academic Press, New York (1954)

    MATH  Google Scholar 

  12. Bonar, D.: On Annular Functions. VEB Deutscher Verlag der Wissenschaften, Berlin (1971)

    MATH  Google Scholar 

  13. Bonar, D., Carroll, F.: Annular functions form a residual set. J. Reine Angew. Math. 272, 23–24 (1975)

    MathSciNet  MATH  Google Scholar 

  14. Bonar, D., Carroll, F., Ramachandra, K.: Not every annular function is strongly annular. J. Reine Angew. Math. 273, 57–60 (1975)

    MathSciNet  MATH  Google Scholar 

  15. Bonar, D., Carroll, F., Erdös, P.: Strongly annular functions with small coefficients and related results. Proc. Am. Math. Soc. 67, 129–132 (1977)

    Article  MATH  Google Scholar 

  16. Bonar, D., Carroll, F., Piranian, G.: Strongly annular functions with small Taylor coefficients. Math. Z. 156, 85–91 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  17. Buckley, S., Koskela, P., Vukotic, D.: Fractional integration, differentiation, and weighted Bergman spaces. Math. Proc. Camb. Philos. Soc. 126, 369–385 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Campbell, D.M., Hwang, J.S.: Annular functions and gap series. Bull. Lond. Math. Soc. 14, 415–418 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Daquila, R.: Strongly annular solutions of Mahler’s functional equation. Complex Var. Theory Appl. 32, 99–104 (1997)

    Article  MathSciNet  Google Scholar 

  20. Daquila, R.: Approximations by strongly annular solutions of functional equations. Proc. Am. Math. Soc. 138, 2505–2511 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. García-Pacheco, F.J., Martín, M., Seoane-Sepúlveda, J.B.: Lineability, spaceability, and algebrability of certain subsets of function spaces. Taiwan. J. Math. 13, 1257–1269 (2009)

    MATH  Google Scholar 

  22. Gurariy, V., Quarta, L.: On lineability of sets of continuous functions. J. Math. Anal. Appl. 294, 62–72 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, New York (2000)

    Book  MATH  Google Scholar 

  24. Holland, A.S.B.: Introduction to the Theory of Entire Functions. Academic Press, New York (1973)

    MATH  Google Scholar 

  25. Howell, R.: Annular functions in probability. Proc. Am. Math. Soc. 52, 217–221 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  26. Osada, A.: On the distribution of zeros of a strongly annular function. Nagoya Math. J. 56, 13–17 (1975)

    MathSciNet  MATH  Google Scholar 

  27. Osada, A.: On the distribution of a-points of a strongly annular function. Pac. J. Math. 68, 491–496 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  28. Osada, A.: Strongly annular functions with given singular values. Math. Scand. 50, 73–78 (1982)

    MathSciNet  MATH  Google Scholar 

  29. Priwalow, I.I.: Randeigenschaften Analytischer Funktionen. VEB Deutscher Verlag der Wissenschaften, Berlin (1971)

    Google Scholar 

  30. Redett, D.: Strongly annular functions in Bergman space. Comput. Methods Funct. Theory 7, 429–432 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Avda. Reina Mercedes, 41080, Sevilla, Spain

    Luis Bernal-González

  2. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de La Laguna, C/Astrofísico Francisco Sánchez, 38271, La Laguna, Tenerife, Spain

    Antonio Bonilla

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  1. Luis Bernal-González
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  2. Antonio Bonilla
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Correspondence to Luis Bernal-González.

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Bernal-González, L., Bonilla, A. Families of strongly annular functions: linear structure. Rev Mat Complut 26, 283–297 (2013). https://doi.org/10.1007/s13163-011-0080-9

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  • DOI: https://doi.org/10.1007/s13163-011-0080-9

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