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This article is cited in 4 scientific papers (total in 4 papers)
MATHEMATICS
Global extrema of the Gray Takagi function of Kobayashi and binary digital sums
O. E. Galkin, S. Yu. Galkina Lobachevsky State University of Nizhni Novgorod,
pr. Gagarina, 23, Nizhni Novgorod, 603950, Russia
Abstract:
The Gray Takagi function $\widetilde{T}(x)$ was defined by Kobayashi in 2002 for calculation of Gray code digital sums. By construction, the Gray Takagi function is similar to the Takagi function, described in 1903. Like the Takagi function, the Gray Takagi function of Kobayashi is continuous, but nowhere differentiable on the real axis. In this paper, we prove that the global maximum for the Gray Takagi function of Kobayashi is equal to $8/15$, and on the segment $[0;2]$ it is reached at those and only those points of the interval $(0;1)$, whose hexadecimal record contains only digits $4$ or $8$. We also show that the global minimum of $\widetilde{T}(x)$ is equal to $-8/15$, and on the segment $[0;2]$ it is reached at those and only those points of the interval $(1;2)$, whose hexadecimal record contains only digits $7$ or $\langle11\rangle$. In addition, we calculate the global minimum of the Gray Takagi function on the segment $[1/2;1]$ and get the value $-2/15$. We find global extrema and extreme points of the function $\log_2 x + \widetilde{T} (x)/x$. By using the results obtained, we get the best estimation of Gray code digital sums from Kobayashi's formula.
Keywords:
continuous nowhere differentiable Gray Takagi function of Kobayashi, global maximum, global extremum, Gray code binary digital sums.
Received: 01.02.2017
Citation:
O. E. Galkin, S. Yu. Galkina, “Global extrema of the Gray Takagi function of Kobayashi and binary digital sums”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 27:1 (2017), 17–25
Linking options:
https://www.mathnet.ru/eng/vuu565 https://www.mathnet.ru/eng/vuu/v27/i1/p17
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