Asymptotic Formula for the Moments of Takagi Function
https://doi.org/10.18255/1818-1015-2016-1-5-11
Abstract
Takagi function is a simple example of a continuous but nowhere differentiable function. It is defined by T(x) = ∞ ᢘ k=0 2−nρ(2nx),
where ρ(x) = min k∈Z |x − k|. The moments of Takagi function are defined as Mn = ᝈ 1 0 xnT(x) dx. The main result of this paper is the following: Mn = lnn − Γ(1) − lnπ n2 ln 2 + 1 2n2 + 2 n2 ln 2 φ(n) + O(n−2.99), where φ(x) = ᝨ kᡘ=0 Γ ᝈ2πik ln 2 ζ ᡸ2πik ln 2 ᡸ x−2lπni2k .
About the Author
E. A. Timofeev
Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Russian Federation
P.G. Demidov
Russian Federation
P.G. Demidov
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Review
For citations:
Timofeev E.A. Asymptotic Formula for the Moments of Takagi Function. Modeling and Analysis of Information Systems. 2016;23(1):5-11. (In Russ.) https://doi.org/10.18255/1818-1015-2016-1-5-11
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