Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 731–735

# 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

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