On global extrema of power Takagi functions

O. E. Galkin ; S. Yu. Galkina ; A. A. Tronov

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 25 (2023) no. 2, pp. 22-36

Résumé

By construction, power Takagi functions $S_p$ are similar to Takagi's continuous nowhere differentiable function described in 1903. These real-valued functions $S_p(x)$ have one real parameter $p > 0$. They are defined on the real axis $\mathbb R$ by the series $S_p(x)=\sum_{n=0}^\infty (S_0(2^nx)/2^n)^p$, where $S_0(x)$ is the distance from real number $x$ to the nearest integer number. We show that for every $p > 0$, the functions $S_p$ are everywhere continuous, but nowhere differentiable on $\mathbb R$. Next, we derive functional equations for Takagi power functions. With these, it is possible, in particular, to calculate the values $S_p(x)$ at rational points $x$. In addition, for all values of the parameter $p$ from the interval $(0;1)$, we find the global extrema of the functions $S_p$, as well as the points where they are reached. It turns out that the global maximum of $S_p$ equals to $2^p/(3^p(2^p-1))$ and is reached only at points $q+1/3$ and $q+2/3$, where $q$ is an arbitrary integer. The global minimum of the functions $S_p$ equals to $0$ and is reached only at integer points. Using the results on global extremes, we obtain two-sided estimates for the functions $S_p$ and find the points at which these estimates are reached.

Détail

Keywords: power Takagi function, continuity, nowhere differentiability, functional equations, global extrema