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Model. Anal. Inform. Sist., 2016, Volume 23, Number 1, Pages 5–11 (Mi mais479)  

Asymptotic formula for the moments of Takagi function

E. A. Timofeev

P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia

Abstract: Takagi function is a simple example of a continuous but nowhere differentiable function. It is defined by
$$ T(x) = \sum_{k=0}^{\infty}2^{-n}\rho(2^nx), $$
where
$$ \rho(x) = \min_{k\in \mathbb{Z}}|x-k|. $$
The moments of Takagi function are defined as
$$ M_n = \int_0^1 x^n T(x) dx. $$
The main result of this paper is the following:
$$ M_n = \frac{\ln n - \Gamma'(1)-\ln\pi}{n^2\ln 2}+\frac{1}{2n^2} +\frac{2}{n^2\ln 2} \phi(n) + \mathcal{O}(n^{-2.99}), $$
where
$$ \phi(x) = \sum_{k\ne 0} \Gamma(\frac{2\pi i k}{\ln 2})\zeta(\frac{2\pi i k}{\ln 2})x^{-\frac{2\pi i k}{\ln 2}}. $$


Keywords: moments, self-similar, Takagi function, singular, Mellin transform, asymptotic.

DOI: https://doi.org/10.18255/1818-1015-2016-1-5-11

Full text: PDF file (540 kB)
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Bibliographic databases:

UDC: 519.17
Received: 20.12.2015

Citation: E. A. Timofeev, “Asymptotic formula for the moments of Takagi function”, Model. Anal. Inform. Sist., 23:1 (2016), 5–11

Citation in format AMSBIB
\Bibitem{Tim16}
\by E.~A.~Timofeev
\paper Asymptotic formula for the moments of Takagi function
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 1
\pages 5--11
\mathnet{http://mi.mathnet.ru/mais479}
\crossref{https://doi.org/10.18255/1818-1015-2016-1-5-11}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3502271}
\elib{http://elibrary.ru/item.asp?id=25475536}


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