О глобальных экстремумах степенных функций Такаги

Журнал Средневолжского математического общества. 2023. Т. 25. № 2. С. 22–36.

Galkin O., Galkina S., Тронов А. А.

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 any $p>0$, the functions $S_p$ on $\mathbb R$ are everywhere continuous, but nowhere differentiable. Next, we derive functional equations for the 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 the functions $S_p$ equals to $2^p/(3^p(2^p-1))$ and is reached only at points of the form $q+1/3$ 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.

Keywords: непрерывность, continuity, global extrema, functional equations, функциональные уравнения, power Takagi function, nowhere differentiability, степенная функция Такаги, нигде не дифференцируемость, глобальные экстремумы