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## Archive of Issues

Russia Nizhni Novgorod
Year
2017
Volume
27
Issue
1
Pages
17-25
 Section Mathematics Title Global extrema of the Gray Takagi function of Kobayashi and binary digital sums Author(-s) Galkin O.E.a, Galkina S.Yu.a Affiliations Nizhni Novgorod State Universitya 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 UDC 517.518 MSC 26A27, 26A06 DOI 10.20537/vm170102 Received 1 February 2017 Language Russian Citation Galkin O.E., Galkina S.Yu. Global extrema of the Gray Takagi function of Kobayashi and binary digital sums, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 1, pp. 17-25. References Allaart P.C., Kawamura K. The Takagi function: a survey, Real Analysis Exchange, 2011, vol. 37, no. 1, pp. 1-54. http://projecteuclid.org/euclid.rae/1335806762 Delange H. Sur la fonction sommatoire de la fonction “Somme des Chiffres”, L'Enseignement Mathéematique, 1975, vol. 21, pp. 31-47. DOI: 10.5169/seals-47328 Galkin O.E., Galkina S.Yu. On properties of functions in exponential Takagi class, Ufa Mathematical Journal, 2015, vol. 7, no. 3, pp. 28-37. DOI: 10.13108/2015-7-3-28 Gray F. Pulse code communication. U.S. Patent no. 2632058, 17 march 1953. http://pdfpiw.uspto.gov/.piw?Docid=02632058 Kahane J.P. Sur l’exemple, donné par M. de Rham, d’une fonction continue sans dérivée, L'Enseignement Mathématique, 1959, vol. 5, pp. 53-57. Kobayashi Z. Digital sum problems for the Gray code representation of natural numbers, Interdisciplinary Information Sciences, 2002, vol. 8, no. 2, pp. 167-175. DOI: 10.4036/iis.2002.167 Krüppel M. Takagi’s continuous nowhere differentiable function and binary digital sums, Rostock. Math. Kolloq., 2008, vol. 63, pp. 37-54. http://ftp.math.uni-rostock.de/pub/romako/heft63/kru63.pdf Lagarias J.C. The Takagi function and its properties, RIMS Kôkyûroku Bessatsu B34: Functions in Number Theory and Their Probabilistic Aspects, Eds.: K. Matsumoto, S. Akiyama, K. Fukuyama, H. Nakada, H. Sugita, A. Tamagawa. Kyoto, 2012, pp. 153-189. http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B34/pdf/B34-11.pdf Martynov B. Van der Waerden’s pathological function, Quantum, 1998, vol. 8, no. 6, pp. 12-19. http://static.nsta.org/pdfs/QuantumV8N6.pdf Takagi T. A simple example of a continuous function without derivative, Tokyo Sugaku-Butsurigakkwai Hokoku, 1901, vol. 1, pp. F176-F177. DOI: 10.11429/subutsuhokoku1901.1.F176 Trollope J.R. An explicit expression for binary digital sums, Mathematics Magazine, 1968, vol. 41, no. 1, pp. 21-25. DOI: 10.2307/2687954 Full text