Abstract
The Takagi function is a simple example of a continuous yet nowhere differentiable function and is given as T(x) = Σk=0∞ 2−n ρ(2nx), where \(\rho (x) = \mathop {\min }\limits_{k \in \mathbb{Z}} |x - k|\). The moments of the Takagi function are given as M n = ∫01xnT(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.
Similar content being viewed by others
References
Flajolet, P. and Sedgewick, R., Analytic Combinatorics, Cambridge University Press, 2008.
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.
Lagarias, J.C., The Takagi function and its properties, RIMS Kôkyûroku Bessatsu, 2012, vol. B34, pp. 153–189.
Allaart, P.C. and Kawamura, K., The Takagi function: A survey, Real Anal. Exchange, 2011, vol. 37, no. 1, pp. 1–54.
De Rham, G., On some curves defined by functional equations, in Classics on Fractals, Edgar, G.A., Ed., 1993, pp. 285–298
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.
Oberhettinger, F., Tables of Mellin Transforms, New York: Springer-Verlag, 1974.
Szpankowski, W., Average Case Analysis of Algorithms on Sequences, New York: John Wiley & Sons, 2001.
Gradstein, I.S. and Ryzhik, I.M., Table of Integrals, Series, and Products, Academic Press, 1994.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.A. Timofeev, 2016, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2016, Vol. 23, No. 1, pp. 5–11.
About this article
Cite this article
Timofeev, E.A. Asymptotic Formula for the Moments of the Takagi Function. Aut. Control Comp. Sci. 51, 731–735 (2017). https://doi.org/10.3103/S0146411617070197
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0146411617070197