Advertisement

Monatshefte für Mathematik

, Volume 188, Issue 4, pp 591–609 | Cite as

Nowhere hölderian functions and Pringsheim singular functions in the disc algebra

  • L. Bernal-González
  • A. Bonilla
  • J. López-Salazar
  • J. B. Seoane-SepúlvedaEmail author
Article

Abstract

We prove the existence of dense linear subspaces, of infinitely generated subalgebras and of infinite dimensional Banach spaces in the disc algebra all of whose nonzero members are not \(\alpha \)-hölderian at any point of the unit circle for any  \(\alpha >0\). This completes the recently established result of topological genericity of this kind of functions, as well as the corresponding lineability statements about functions that are nowhere differentiable at the boundary. Topological and algebraic genericity is also studied for the family of boundary-smooth holomorphic functions that are Pringsheim singular at any point of the unit circle.

Keywords

Nowhere hölderian function Pringsheim singular function Disc algebra Lineability Spaceability Algebrability 

Mathematics Subject Classification

Primary 30H50 Secondary 15A03 26A27 26E10 46E10 

Notes

Acknowledgements

The first author was supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P. The second author was supported by MINECO Project MTM2016-75963-P. The third and fourth authors were supported by MEC Grant MTM2015-65825-P.

References

  1. 1.
    Ahlfors, L.V.: Complex Analysis, 3rd edn. McGraw-Hill, London (1979)zbMATHGoogle Scholar
  2. 2.
    Aizpuru, A., Pérez-Eslava, C., Seoane-Sepúlveda, J.B.: Linear structure of sets of divergent sequences and series. Linear Algebra Appl. 418(2–3), 595–598 (2006).  https://doi.org/10.1016/j.laa.2006.02.041 MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aron, R.M., Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lineability: the search for linearity in mathematics, Monographs and Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton, FL (2016)Google Scholar
  4. 4.
    Aron, R.M., García-Pacheco, F.J., Pérez-García, D., Seoane-Sepúlveda, J.B.: On dense-lineability of sets of functions on \({\mathbb{R}}\). Topology 48(2–4), 149–156 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on \({\mathbb{R}}\). Proc. Am. Math. Soc. 133(3), 795–803 (2005).  https://doi.org/10.1090/S0002-9939-04-07533-1 MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Aron, R.M., Pérez-García, D., Seoane-Sepúlveda, J.B.: Algebrability of the set of nonconvergent Fourier series. Stud. Math. 175(1), 83–90 (2006)zbMATHCrossRefGoogle Scholar
  7. 7.
    Auerbach, H., Banach, S.: Über die Höldersche Bedingung. Stud. Math. 3, 180–184 (1931). (in German)zbMATHCrossRefGoogle Scholar
  8. 8.
    Balcerzak, M., Bartoszewicz, A., Filipczak, M.: Nonseparable spaceability and strong algebrability of sets of continuous singular functions. J. Math. Anal. Appl. 407(2), 263–269 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Banach, S.: Über die Bairesche Kategorie gewisser Funktionenmengen. Stud. Math. 3, 174–179 (1931). (in German)zbMATHCrossRefGoogle Scholar
  10. 10.
    Bartoszewicz, A., Bienias, M., Filipczak, M., Gła̧b, S.: Strong \({\mathfrak{c}}\)-algebrability of strong Sierpiński–Zygmund, smooth nowhere analytic and other sets of functions. J. Math. Anal. Appl. 412(2), 620–630 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bartoszewicz, A., Gła̧b, S.: Large function algebras with certain topological properties. J. Funct. Spaces 2015, 7 (2015)Google Scholar
  12. 12.
    Bastin, F., Conejero, J.A., Esser, C., Seoane-Sepúlveda, J.B.: Algebrability and nowhere Gevrey differentiability. Israel J. Math. 205(1), 127–143 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bayart, F., Quarta, L.: Algebras in sets of queer functions. Israel J. Math. 158(1), 285–296 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bernal-González, L.: Funciones con derivadas sucesivas grandes y pequeñas por doquier. Collect. Math. 38, 117–122 (1987). (in Spanish)MathSciNetGoogle Scholar
  15. 15.
    Bernal-González, L.: Linear Kierst–Szpilrajn theorems. Stud. Math. 166(1), 55–69 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bernal-González, L.: Lineability of sets of nowhere analytic functions. J. Math. Anal. Appl. 340(2), 1284–1295 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Bernal-González, L.: Algebraic genericity of strict-order integrability. Stud. Math. 199(3), 279–293 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Bernal-González, L.: Vector spaces of non-extendable holomorphic functions. J. Anal. Math., to appearGoogle Scholar
  19. 19.
    Bernal-González, L., Calderón-Moreno, M.C.: Large algebras of singular functions vanishing on prescribed sets. Results Math. 71(3), 1207–1221 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Bernal-González, L., Conejero, J. A., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Highly tempering infinite matrices. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 112(2), 341–345 (2018).  https://doi.org/10.1007/s13398-017-0385-8.MR3775272
  21. 21.
    Bernal-González, L., López-Salazar, J., Seoane, J.B.: On Weierstrass’ monsters in the disc algebra. Bull. Belg. Math. Soc, Simon Stevin (2018). in pressGoogle Scholar
  22. 22.
    Bernal-González, L., Ordóñez Cabrera, M.: Lineability criteria, with applications. J. Funct. Anal. 266(6), 3997–4025 (2014).  https://doi.org/10.1016/j.jfa.2013.11.014 MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. (N.S.) 51(1), 71–130 (2014).  https://doi.org/10.1090/S0273-0979-2013-01421-6 MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Cariello, D., Fávaro, V.V., Seoane-Sepúlveda, J.B.: Self-similar functions, fractals and algebraic genericity. Proc. Am. Math. Soc. 145(10), 4151–4159 (2017).  https://doi.org/10.1090/proc/13552 MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Cariello, D., Seoane-Sepúlveda, J.B.: Basic sequences and spaceability in \(\ell _p\) spaces. J. Funct. Anal. 266(6), 3797–3814 (2014).  https://doi.org/10.1016/j.jfa.2013.12.011 MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Cartan, H.: Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, language=in French, 6th edn. Hermann, Paris (1997)Google Scholar
  27. 27.
    Cater, F.S.: Differentiable, nowhere analytic functions. Am. Math. Mon. 91(10), 618–624 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Ciesielski, K.C., Gámez-Merino, J.L., Natkaniec, T., Seoane-Sepúlveda, J.B.: On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity. Topol. Appl. 235, 73–82 (2018).  https://doi.org/10.1016/j.topol.2017.12.017 MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Ciesielski, K.C., Gámez-Merino, J.L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lineability, spaceability, and additivity cardinals for Darboux-like functions. Linear Algebra Appl. 440, 307–317 (2014).  https://doi.org/10.1016/j.laa.2013.10.033 MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Craik, A.D.D.: Prehistory of Faà di Bruno’s Formula. Am. Math. Mon. 112(2), 217–234 (2005)zbMATHGoogle Scholar
  31. 31.
    Darst, R.B.: Most infinitely differentiable functions are nowhere analytic. Canad. Math. Bull. 16, 597–598 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Eskenazis, A.: Topological genericity of nowhere differentiable functions in the disc algebra. Arch. Math. (Basel) 103(1), 85–92 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Eskenazis, A., Makridis, K.: Topological genericity of nowhere differentiable functions in the disc and polydisc algebras. J. Math. Anal. Appl. 420(1), 435–446 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Esser, C.: Generic results in classes of ultradifferentiable functions. J. Math. Anal. Appl. 413(1), 378–391 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Fonf, V.P., Gurariy, V.I., Kadets, M.I.: An infinite dimensional subspace of \(C[0,1]\) consisting of nowhere differentiable functions. C. R. Acad. Bulgare Sci. 52(11–12), 13–16 (1999)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Gámez, J.L., Muñoz-Fernández, G.A., Seoane-Sepúlveda, J.B.: Lineability and additivity in \({\mathbb{R}^{\mathbb{R}}}\). J. Math. Anal. Appl. 369(1), 265–272 (2010).  https://doi.org/10.1016/j.jmaa.2010.03.036 MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Gámez-Merino, J.L., Muñoz-Fernández, G.A., Seoane-Sepúlveda, J.B.: A characterization of continuity revisited. Am. Math. Mon. 118(2), 167–170 (2011).  https://doi.org/10.4169/amer.math.monthly.118.02.167 MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Gámez-Merino, J.L., Seoane-Sepúlveda, J.B.: An undecidable case of lineability in \({\mathbb{R}^{\mathbb{R}}}\). J. Math. Anal. Appl. 401(2), 959–962 (2013).  https://doi.org/10.1016/j.jmaa.2012.10.067 MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    García, D., Grecu, B.C., Maestre, M., Seoane-Sepúlveda, J.B.: Infinite dimensional Banach spaces of functions with nonlinear properties. Math. Nachr. 283(5), 712–720 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Gurariĭ, V.I.: Subspaces and bases in spaces of continuous functions. Dokl. Akad. Nauk SSSR 167, 971–973 (1966). (Russian)MathSciNetGoogle Scholar
  41. 41.
    Gurariĭ, V.I.: Linear spaces composed of everywhere nondifferentiable functions. C. R. Acad. Bulgare Sci. 44(5), 13–16 (1991). (in Russian)MathSciNetGoogle Scholar
  42. 42.
    Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Stud. Math. 184, 233–247 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Hencl, S.: Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Proc. Am. Math. Soc. 128(12), 3505–3511 (2000)zbMATHCrossRefGoogle Scholar
  44. 44.
    Hunt, B.R.: The prevalence of continuous nowhere differential functions. Proc. Am. Math. Soc. 122(3), 711–717 (1994)zbMATHCrossRefGoogle Scholar
  45. 45.
    Jiménez-Rodríguez, P., Muñoz-Fernández, G.A., Seoane-Sepúlveda, J.B.: On Weierstrass’ Monsters and lineability. Bull. Belg. Math. Soc. Simon Stevin 20(4), 577–586 (2013)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Kavvadias, K., Makridis, K.: Nowhere differentiable functions with respect to the position. Preprint (2017). arXiv:1701.04875v1 [math.CV]
  47. 47.
    Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Mazurkiewicz, S.: Sur les fonctions non dérivables. Stud. Math. 3, 92–94 (1931)zbMATHCrossRefGoogle Scholar
  49. 49.
    Morgenstern, D.: Unendlich oft differenzierbare nicht-analytische Funktionen. Math. Nachr. 12, 74 (1954). (in German)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Oxtoby, J.C.: Measure and Category, 2nd edn. Springer, New York (1980)zbMATHCrossRefGoogle Scholar
  51. 51.
    Ramsamujh, T.I.: Nowhere analytic \(C^\infty \) functions. J. Math. Anal. Appl. 160, 263–266 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Rodríguez-Piazza, L.: Every separable Banach space is isometric to a space of continuous nowhere differentiable functions. Proc. Am. Math. Soc. 123(12), 3649–3654 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)zbMATHGoogle Scholar
  54. 54.
    Salzmann, H., Zeller, K.: Singularitäten unendlicht oft differenzierbarer Funktionen. Math. Z. 62, 354–367 (1955). (in German)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Siciak, J.: Regular and singular points of \(C^\infty \) functions. Zesz. Nauk. Politech. Sl., Math.-Fiz 48(853), 127–146 (1986). (in Polish)zbMATHGoogle Scholar
  56. 56.
    Seoane-Sepúlveda, J.B.: Chaos and lineability of pathological phenomena in analysis. Thesis (Ph.D.)–Kent State University, ProQuest LLC, Ann Arbor, MI, 2006, 139, isbn=978-0542-78798-0Google Scholar
  57. 57.
    Weierstrass, K.: Über continuirliche Funktionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen, Gelesen Akad. Wiss, 18 Juli 1872 (in German)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • L. Bernal-González
    • 1
  • A. Bonilla
    • 2
  • J. López-Salazar
    • 3
  • J. B. Seoane-Sepúlveda
    • 4
    Email author
  1. 1.Departamento de Análisis Matemático Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Departamento de Análisis MatemáticoUniversidad de la LagunaLa LagunaSpain
  3. 3.Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones Escuela Técnica Superior de Ingeniería y Sistemas de TelecomunicaciónUniversidad Politécnica de MadridMadridSpain
  4. 4.Instituto de Matemática Interdisciplinar (IMI), Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain

Personalised recommendations