Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 731–735 | Cite as

Asymptotic Formula for the Moments of the Takagi Function

Article

Abstract

The Takagi function is a simple example of a continuous yet nowhere differentiable function and is given as T(x) = Σk=0 2n ρ(2 n x), where \(\rho (x) = \mathop {\min }\limits_{k \in \mathbb{Z}} |x - k|\). The moments of the Takagi function are given as M n = ∫01x n T(x)dx. The estimate \({M_n} = \frac{{1nn - \Gamma '(1) - 1n\pi }}{{{n^2}1n2}} + \frac{1}{{2{n^2}}} + \frac{2}{{{n^2}1n2}}\phi (n) + O({n^{ - 2.99}})\), where the function \(\phi (x) = \sum\nolimits_{k \ne 0} \Gamma (\frac{{2\pi ik}}{{1n2}})\zeta (\frac{{2\pi ik}}{{1n2}}){x^{ - \frac{{2\pi ik}}{{1n2}}}}\) is periodic in log2x and Γ(x) and ζ(x) denote the gamma and zeta functions, is the principal result of this work.

Keywords

moments self-similarity Takagi function singular function Mellin transform asymptotic formula 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Flajolet, P. and Sedgewick, R., Analytic Combinatorics, Cambridge University Press, 2008.MATHGoogle Scholar
  2. 2.
    Flajolet, P., Gourdon, X., and Dumas, P., Mellin transforms and asymptotics: Harmonic sums, Theor. Comput. Sci., 1995, vol. 144, nos. 1–2, pp. 3–58.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Lagarias, J.C., The Takagi function and its properties, RIMS Kôkyûroku Bessatsu, 2012, vol. B34, pp. 153–189.MathSciNetMATHGoogle Scholar
  4. 4.
    Allaart, P.C. and Kawamura, K., The Takagi function: A survey, Real Anal. Exchange, 2011, vol. 37, no. 1, pp. 1–54.MathSciNetMATHGoogle Scholar
  5. 5.
    De Rham, G., On some curves defined by functional equations, in Classics on Fractals, Edgar, G.A., Ed., 1993, pp. 285–298Google Scholar
  6. 6.
    Kairies, H.-H., Darsow, W.F., and Frank, M.J., Functional equations for a function of van der Waerden type, Rad. Mat., 1988, vol. 4, no. 2, pp. 361–374.MathSciNetMATHGoogle Scholar
  7. 7.
    Oberhettinger, F., Tables of Mellin Transforms, New York: Springer-Verlag, 1974.CrossRefMATHGoogle Scholar
  8. 8.
    Szpankowski, W., Average Case Analysis of Algorithms on Sequences, New York: John Wiley & Sons, 2001.CrossRefMATHGoogle Scholar
  9. 9.
    Gradstein, I.S. and Ryzhik, I.M., Table of Integrals, Series, and Products, Academic Press, 1994.Google Scholar

Copyright information

© Allerton Press, Inc. 2017

Authors and Affiliations

  1. 1.Demidov Yaroslavl State UniversityYaroslavlRussia

Personalised recommendations