Geodesic
Global extrema of the Gray Takagi function of Kobayashi and binary digital sums
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 1, pp. 17-25 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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O. E. Galkin; S. Yu. Galkina. Global extrema of the Gray Takagi function of Kobayashi and binary digital sums. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 1, pp. 17-25. http://geodesic.mathdoc.fr/item/VUU_2017_27_1_a1/

[1] Allaart P. C., Kawamura K., “The Takagi function: a survey”, Real Analysis Exchange, 37:1 (2011), 1–54 http://projecteuclid.org/euclid.rae/1335806762 | DOI | MR

[2] Delange H., “Sur la fonction sommatoire de la fonction “Somme des Chiffres””, L'Enseignement Mathématique, 21 (1975), 31–47 | DOI | MR | Zbl

[3] Galkin O. E., Galkina S. Yu., “On properties of functions in exponential Takagi class”, Ufa Mathematical Journal, 7:3 (2015), 28–37 | DOI | MR

[4] Gray F., Pulse code communication, U.S. Patent No 2632058, , 17 march 1953 http://pdfpiw.uspto.gov/.piw?Docid=02632058

[5] Kahane J. P., “Sur l'exemple, donné par M. de Rham, d'une fonction continue sans dérivée”, L'Enseignement Mathématique, 5 (1959), 53–57 | MR | Zbl

[6] Kobayashi Z., “Digital sum problems for the Gray code representation of natural numbers”, Interdisciplinary Information Sciences, 8:2 (2002), 167–175 | DOI | MR | Zbl

[7] Krüppel M., “Takagi's continuous nowhere differentiable function and binary digital sums”, Rostock. Math. Kolloq., 63 (2008), 37–54 http://ftp.math.uni-rostock.de/pub/romako/heft63/kru63.pdf | MR | Zbl

[8] Lagarias J. C., “The Takagi function and its properties”, Functions in Number Theory and Their Probabilistic Aspects (Kyoto, 2012), RIMS Kôkyûroku Bessatsu, B34, eds. K. Matsumoto, S. Akiyama, K. Fukuyama, H. Nakada, H. Sugita, A. Tamagawa, 153–189 http://www.kurims.kyoto-u.ac.jp/k̃enkyubu/bessatsu/open/B34/pdf/B34-11.pdf | MR | Zbl

[9] Martynov B., “Van der Waerden's pathological function”, Quantum, 8:6 (1998), 12–19 http://static.nsta.org/pdfs/QuantumV8N6.pdf | MR

[10] Takagi T., “A simple example of a continuous function without derivative”, Tokyo Sugaku–Butsurigakkwai Hokoku, 1 (1901), F176–F177 | DOI

[11] Trollope J. R., “An explicit expression for binary digital sums”, Mathematics Magazine, 41:1 (1968), 21–25 | DOI | MR | Zbl