On global extrema of power Takagi functions

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Abstract

By construction, power Takagi functions Sp are similar to Takagi's continuous nowhere differentiable function described in 1903. These real-valued functions Sp have one real parameter p>0. They are defined on the real axis RR"> by the series Sp(x)=∑∞n=0 (S0(2nx)/2n)p, where S0(x) is the distance from real number x to the nearest integer number. We show that for every p>0, the functions Sp are everywhere continuous, but nowhere differentiable on R. Next, we derive functional equations for Takagi power functions. With these, it is possible, in particular, to calculate the values Sp(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 Sp, as well as the points where they are reached. It turns out that the global maximum of Sp equals to 2p/(3p(2p-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 Sp equals to 0 and is reached only at integer points. Using the results on global extremes, we obtain two-sided estimates for the functions Sp and find the points at which these estimates are reached.

About the authors

Oleg E. Galkin

National Research University «Higher School of Economics»

Email: olegegalkin@ya.ru
ORCID iD: 0000-0003-2085-572X

Ph.D. (Phys.-Math.), Associate Professor, Department of Fundamental Mathematics

Russian Federation, 25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia

Svetlana Yu. Galkina

National Research University «Higher School of Economics»

Email: svetlana.u.galkina@mail.ru
ORCID iD: 0000-0002-2476-2275

Ph.D. (Phys.-Math.), Associate Professor, Department of Fundamental Mathematics

Russian Federation, 25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia

Anton A. Tronov

National Research University «Higher School of Economics»

Author for correspondence.
Email: tronovaa@yandex.ru
ORCID iD: 0009-0000-6454-1226

master’s student of the Faculty of Informatics, Mathematics and Computer Science

Russian Federation, 25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia

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Copyright (c) 2023 Galkin O.E., Galkina S.Y., Tronov A.A.

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