Acta Mathematica Hungarica

, Volume 151, Issue 2, pp 361–378 | Cite as

A Salem generalised function

  • E. de Amo
  • M. Díaz Carrillo
  • J. Fernández-Sánchez
Article

Abstract

Among the members of the celebrated family of functions introduced by Salem in the mid 20th century, there is a particular and very interesting one that we use to relate the dyadic system of numbers representation with the modified Engel system. Various properties are studied for this function, including derivatives and fractal dimensions.

Key words and phrases

singular function Engel representation asymptotic distribution function Hausdorff (fractal) dimension 

Mathematics Subject Classification

26A30 26A24 

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References

  1. 1.
    Cantor G.: De la puissance des ensembles parfait de points. Acta Math. 4, 381–392 (1884)MathSciNetCrossRefGoogle Scholar
  2. 2.
    H. Lebesgue, Leçons sur l’integration et la Recherche des Fonctions Primitives, Gauthiers-Villars (Paris, 1904).Google Scholar
  3. 3.
    P. Billingsley, Probability and Measure (2nd ed.), Wiley (New York, 1995).Google Scholar
  4. 4.
    H. Minkowski, Verhandlungen des III Internationalen Mathematiker-kongresses in Heidelberg (1904).Google Scholar
  5. 5.
    Viader P., Paradís J., Bibiloni L.: A new light on Minkowski’s ?(x) function. J. Number Theory 73, 212–227 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Paradís J., Viader P., Bibiloni L.: On actually computable bijections between \({\mathbb{N}}\) and \({\mathbb{Q}^{+}}\). Order 13, 369–377 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Paradís J., Viader P., Bibiloni L.: A total order in (0,1] defined through a ‘Next’ Operator. Order 16, 207–220 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    E. de Amo, M. Díaz Carrillo and J. Fernández-Sánchez, Harmonic analysis on the Sierpinski gasket and singular functions, Acta Math. Hungar., 143 (2013), 58–74.Google Scholar
  9. 9.
    Bohnstengel J., Kesseböhmer M.: Wavelets for iterated function systems. J. Funct. Anal. 259, 583–601 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    E. de Amo, M. Díaz Carrillo and J. Fernández-Sánchez, Singular functions with applications to fractals and generalised Takagi’s functions, Acta Appl. Math., 119 (2012), 129–148.Google Scholar
  11. 11.
    E. de Amo, M. Díaz Carrillo and J. Fernández-Sánchez, On duality of aggregation operators and k-negations, Fuzzy Sets and Systems, 181 (2011), 14–27.Google Scholar
  12. 12.
    Salem R.: On some singular monotone functions which are strictly increasing. Trans. Amer. Math. Soc. 53, 427–439 (1943)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Paradís J., Viader P., Bibiloni L.: Riesz–Nagy singular functions revisited. J. Math. Anal. Appl. 329, 592–602 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Takács L.: An increasing continuous singular function. Amer. Math. Soc. 85, 35–36 (1978)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rényi A.: A new approach to the theory of Engel’s series. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 5, 25–32 (1962)MathSciNetMATHGoogle Scholar
  16. 16.
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (4th ed.), England Clarendon Press (Oxford, 1979).Google Scholar
  17. 17.
    W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co. (New York, 1970).Google Scholar
  18. 18.
    K. J. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley&Sons (Chichester, 1990).Google Scholar
  19. 19.
    Berg L., Kruppel M.: De Rham’s singular function and related functions. Z. Anal. Anwend. 19, 227–237 (2000)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    E. Hewitt and K. R. Stromberg, Real and Abstract Analysis, Springer-Verlag (New York, 1965).Google Scholar
  21. 21.
    Kairies H. H.: Functional equations for peculiar functions. Aequationes Math. 53, 207–241 (1997)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Wen L.: An approach to construct the singular monotone functions by using Markov chains. Taiwanese J. Math. 2, 361–368 (1998)MathSciNetMATHGoogle Scholar
  23. 23.
    E. de Amo and J. FernándezSánchez, A generalised dyadic representation system, Int. J. Pure Appl. Math., 52 (2009), 49–66.Google Scholar
  24. 24.
    G. de Rham, Sur quelques courbes definies par des aequations fonctionnelles, Univ. Politec. Torino. Rend. Sem. Mat., 16 (1956), 101–113.Google Scholar
  25. 25.
    Erdős P.: On the smoothness of the asymptotic distribution of additive arithmetical functions. Amer. J. Math. 61, 722–725 (1939)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    H. Niederreiter and L. Kuipers, Uniform Distribution of Sequences, John Wiley&Sons (New York, 1974).Google Scholar
  27. 27.
    J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer (Berlin, 1976).Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  • E. de Amo
    • 1
  • M. Díaz Carrillo
    • 2
  • J. Fernández-Sánchez
    • 1
  1. 1.Universidad de AlmeríaAlmeríaSpain
  2. 2.Universidad de GranadaGranadaSpain

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