Skip to main content
SpringerLink
Log in

A Salem generalised function

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cantor G.: De la puissance des ensembles parfait de points. Acta Math. 4, 381–392 (1884)

    Article  MathSciNet  Google Scholar 

  2. H. Lebesgue, Leçons sur l’integration et la Recherche des Fonctions Primitives, Gauthiers-Villars (Paris, 1904).

  3. P. Billingsley, Probability and Measure (2nd ed.), Wiley (New York, 1995).

  4. H. Minkowski, Verhandlungen des III Internationalen Mathematiker-kongresses in Heidelberg (1904).

  5. Viader P., Paradís J., Bibiloni L.: A new light on Minkowski’s ?(x) function. J. Number Theory 73, 212–227 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Paradís J., Viader P., Bibiloni L.: On actually computable bijections between \({\mathbb{N}}\) and \({\mathbb{Q}^{+}}\). Order 13, 369–377 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Paradís J., Viader P., Bibiloni L.: A total order in (0,1] defined through a ‘Next’ Operator. Order 16, 207–220 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  9. Bohnstengel J., Kesseböhmer M.: Wavelets for iterated function systems. J. Funct. Anal. 259, 583–601 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

  12. Salem R.: On some singular monotone functions which are strictly increasing. Trans. Amer. Math. Soc. 53, 427–439 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  13. Paradís J., Viader P., Bibiloni L.: Riesz–Nagy singular functions revisited. J. Math. Anal. Appl. 329, 592–602 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Takács L.: An increasing continuous singular function. Amer. Math. Soc. 85, 35–36 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rényi A.: A new approach to the theory of Engel’s series. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 5, 25–32 (1962)

    MathSciNet  MATH  Google Scholar 

  16. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (4th ed.), England Clarendon Press (Oxford, 1979).

  17. W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co. (New York, 1970).

  18. K. J. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley&Sons (Chichester, 1990).

  19. Berg L., Kruppel M.: De Rham’s singular function and related functions. Z. Anal. Anwend. 19, 227–237 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Hewitt and K. R. Stromberg, Real and Abstract Analysis, Springer-Verlag (New York, 1965).

  21. Kairies H. H.: Functional equations for peculiar functions. Aequationes Math. 53, 207–241 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wen L.: An approach to construct the singular monotone functions by using Markov chains. Taiwanese J. Math. 2, 361–368 (1998)

    MathSciNet  MATH  Google Scholar 

  23. E. de Amo and J. FernándezSánchez, A generalised dyadic representation system, Int. J. Pure Appl. Math., 52 (2009), 49–66.

  24. G. de Rham, Sur quelques courbes definies par des aequations fonctionnelles, Univ. Politec. Torino. Rend. Sem. Mat., 16 (1956), 101–113.

  25. Erdős P.: On the smoothness of the asymptotic distribution of additive arithmetical functions. Amer. J. Math. 61, 722–725 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Niederreiter and L. Kuipers, Uniform Distribution of Sequences, John Wiley&Sons (New York, 1974).

  27. J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer (Berlin, 1976).

Download references

Author information

Authors and Affiliations

  1. Universidad de Almería, 04120, Almería, Spain

    E. de Amo & J. Fernández-Sánchez

  2. Universidad de Granada, 18071, Granada, Spain

    M. Díaz Carrillo

Authors
  1. E. de Amo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Díaz Carrillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Fernández-Sánchez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. de Amo.

Additional information

This work was supported by the Ministerio de Economía y Competitividad (Spain) under Research Project No. MTM2014-60594-P and partially supported by FEDER.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Amo, E., Carrillo, M.D. & Fernández-Sánchez, J. A Salem generalised function. Acta Math. Hungar. 151, 361–378 (2017). https://doi.org/10.1007/s10474-017-0690-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-017-0690-x

Key words and phrases

Mathematics Subject Classification

Search

Navigation