Advertisement

Ukrainian Mathematical Journal

, Volume 65, Issue 3, pp 448–462 | Cite as

Self-Affine Singular and Nowhere Monotone Functions Related to the Q-Representation of Real Numbers

  • M. V. Prats’ovytyi
  • A.V. Kalashnikov
Article

We study functional, differential, integral, self-affine, and fractal properties of continuous functions from a finite-parameter family of functions with a continual set of “peculiarities.” Almost all functions in this family are singular (their derivative is equal to zero almost everywhere in a sense of the Lebesgue measure) or nowhere monotone and, in particular, not differentiable. We consider various approaches to the definition of these functions (by using a system of functional equations, projectors of the symbols of various representations, distributions of random variables, etc.).

Keywords

Monotone Function Ukrainian National Academy Singular Function Bernoulli Convolution Singular Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. F. Turbin and N. V. Pratsevityi, Fractal Sets, Functions, and Distributions [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  2. 2.
    R. Salem, “On some singular monotonic functions which are strictly increasing,” Trans. Amer. Math. Soc., 423–439 (1943).Google Scholar
  3. 3.
    B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco (1964).zbMATHGoogle Scholar
  4. 4.
    W. Sierpiński, “Elementary example of an increasing function whose derivative is equal to zero almost everywhere,” Mat. Sb., 30, Issue 3 (1916).Google Scholar
  5. 5.
    M. V. Prats’ovytyi, Fractal Approach to the Investigation of Singular Distributions [in Ukrainian], Drahomanov National Pedagogic University, Kyiv (1998).Google Scholar
  6. 6.
    G. Marsalia, “Random variables with independent binary digits,” Ann. Math. Statist., 42, No. 2, 1922–1929 (1971).MathSciNetCrossRefGoogle Scholar
  7. 7.
    H. Minkowski, Gesammeine Abhandlungen, Bd. 2, Berlin (1911).Google Scholar
  8. 8.
    T. Takagi, “A simple example of the continuous function without derivative,” Proc. Phys. Math. Soc. Jpn., 1, 176–177 (1903).Google Scholar
  9. 9.
    S. Albeverio, Ya. Goncharenco, M. Pratsiovytyi, and G. Torbin, “Convolutions of distributions of random variables with independent binary digits,” Random Operators Stochast. Equat., 15, No. 1, 89–97 (2007).CrossRefzbMATHGoogle Scholar
  10. 10.
    S. D. Chatterji, “Certain induced measures on the unit interval,” J. London Math. Soc., 38, 325–331 (1963).MathSciNetCrossRefGoogle Scholar
  11. 11.
    H. Lebesgue, Leçons sur l’Intêgration et la Recherche des Fonctions Primitives, Gauthier-Villars, Paris (1928).zbMATHGoogle Scholar
  12. 12.
    T. Zamfirescu, “Most monotone functions are singular,” Amer. Math. Mon., 88, 47–49 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. V. Prats’ovytyi and H. M. Torbin, “Fractal geometry and transformations preserving the Hausdorff–Besicovitch dimension,” in: Proc. of the Ukrainian Mathematical Congress, “Dynamical Systems” [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2003), pp. 77–93.Google Scholar
  14. 14.
    S. Albeverio, M. Pratsiovytyi, and G. Torbin, “Fractal probability distributions and transformations preserving the Hausdorff–Besicovitch dimension,” Ergod. Theory Dynam. Systems, 24, 1–16 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. V. Prats’ovytyi, “Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction,” Ukr. Mat. Zh., 48, No. 8, 1086–1095 (1996); English translation: Ukr. Math. J., 48, No. 8, 1229–1240 (1996).CrossRefGoogle Scholar
  16. 16.
    I. M. V. Prats’ovyta, “On expansions of numbers in alternating s-adic series and Ostrogradskii series of the first and second kinds,” Ukr. Mat. Zh., 61, No. 7, 958–968 (2009); English translation: Ukr. Math. J., 61, No. 7, 1137–1150 (2009).CrossRefGoogle Scholar
  17. 17.
    S. B. Kozyrev, “On the topological density of winding functions,” Mat. Zametki, 33, No. 1, 71–76 (1983).MathSciNetzbMATHGoogle Scholar
  18. 18.
    M. V. Prats’ovytyi and A.V. Kalashnikov, “On one class of continuous functions with complex local structure most of which are singular or not differentiable,” in: Proc. of the Ukrainian Mathematical Congress, “Dynamical Systems”, Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, 23 (2011), pp. 178–189.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • M. V. Prats’ovytyi
    • 1
  • A.V. Kalashnikov
    • 1
  1. 1.Institute of Mathematics, Ukrainian National Academy of SciencesKyivUkraine

Personalised recommendations