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

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. 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.

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

For citation: O. E. Galkin, S. Yu. Galkina, A. A. Tronov. On global extrema of power Takagi functions. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:2(2023), 22–36. DOI: https://doi.org/10.15507/2079-6900.25.202302.22-36

Information about the authors:

Oleg E. Galkin, Associate Professor, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia), Ph.D. (Phys.-Math.), ORCID: https://orcid.org/0000-0003-2085-572X, olegegalkin@ya.ru

Svetlana Yu. Galkina, Associate Professor, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia), Ph.D. (Phys.-Math.), ORCID: http://orcid.org/0000-0002-2476-2275, svetlana.u.galkina@mail.ru

Anton A. Tronov, master’s student of the Faculty of Informatics, Mathematics and Computer Science, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: http://orcid.org/0009-0000-6454-1226, tronovaa@yandex.ru