Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers

Автор(и)

  • S.O. Serbenyuk Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereschenkivska St., Kyiv, 01004, Ukraine

DOI:

https://doi.org/10.15407/mag14.02.197

Ключові слова:

нiде недиференцiйовнi функцiї, s-адичнi представлення, нега-s-адичнi представлення, немонотоннi функцiї, розмiрнiсть Гаусдорфа-Безiковича.

Анотація

Цю роботу присвячено деякому пiдкласу недиференцiйовних функцiй. Аргументи i значення функцiй, що розглядаються, подано через s-ве або нега-s-ве зображення дiйсних чисел. Технiка моделювання таких функцiй є простiшою в порiвняннi з добре вiдомими технiками моделювання недиференцiйовних функцiй. Iншими словами, значення цих функцiй отримано з s-го або нега-s-го зображення аргументу за допомоги певної замiни цифр чи комбiнацiй цифр.
Описано iнтегральнi, фрактальнi та iншi властивостi розглянутих функцiй.

2010: 26A27, 11B34, 11K55, 39B22.

Посилання

V.F. Brzhechka, On the Bolzano function, Uspekhi Mat. Nauk 4 (1949), 15–21 (Russian).

E. Kel’man, Bernard Bolzano, Izd-vo AN SSSR, Moscow, 1955 (Russian).

G.H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), 301–325. https://doi.org/10.2307/1989005

J. Gerver, More on the differentiability of the Rieman function, Amer. J. Math. 93 (1971), 33–41. https://doi.org/10.2307/2373445

P. Du Bois-Reymond, Versuch einer Classification der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen, J. Reine Angew. Math. 79 (1875), 21–37 (German).

G. Darboux, Mémoire sur les fonctions discontinues, Ann. Sci. École Norm. Sup. 4 (1875), 57–112 (French).

G. Darboux, Addition au mémoire sur les fonctions discontinues, Ann. Sci. École Norm. Sup. 8 (1879), 195–202 (French).

W. Orlicz, Sur les fonctions continues non dérivables, Fund. Math. 34 (1947), 45–60 (French). https://doi.org/10.4064/fm-34-1-45-60

U. Dini, Fondamenti per la teoretica delle funzioni de variabili reali, Tipografia T. Nistri e C., Pisa, 1878 (Italian).

H. Hankel, Untersuchungen über die unendlich oft oscillirenden und unstetigen Functionen, Ludwig Friedrich Fues, Tübingen, 1870 (German).

S. Banach, Uber die Baire’sche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174–179 (German). https://doi.org/10.4064/sm-3-1-174-179

A.S. Besicovitch, Investigation of continuous functions in connection with the question of their differentiability, Mat. Sb. 31 (1924), 529–556 (Russian).

S. Mazurkiewicz, Sur les fonctions non dérivables, Studia Math. 3 (1931), 92–94 (French). https://doi.org/10.4064/sm-3-1-92-94

S. Saks, On the functions of Besicovitch in the space of continuous functions, Fund. Math. 19 (1932), 211–219. https://doi.org/10.4064/fm-19-1-211-219

K.A. Bush, Continuous functions without derivatives, Amer. Math. Monthly 59 (1952), 222–225. https://doi.org/10.2307/2308221

G. Cantor, Ueber die einfachen Zahlensysteme, Z. Math. Phys. 14 (1869), 121–128 (German).

R. Salem, On some singular monotonic functions which are stricly increasing, Trans. Amer. Math. Soc. 53 (1943), 423–439. https://doi.org/10.1090/S0002-9947-1943-0007929-6

S.O. Serbenyuk, On one nearly everywhere continuous and nowhere differentiable function, that defined by automaton with finite memory, Naukovyi Chasopys NPU im. M.P. Dragomanova. Ser. 1. Phizyko-matematychni Nauky 13 (2012), 166–182 (Ukrainian). Available from: https://www.researchgate.net/publication/292970012

S.O. Serbenyuk, On one nearly everywhere continuous and nowhere differentiable function defined by automaton with finite memory, conference abstract (2012) (Ukrainian). Available from: https://www.researchgate.net/publication/311665377

S.O. Serbenyuk, On one nearly everywhere continuous and almost nowhere differentiable function, that defined by automaton with finite memory and preserves the Hausdorff-Besicovitch dimension, preprint (2012) (Ukrainian). Available from: https://www.researchgate.net/publication/314409844

S.O. Serbenyuk, On one generalization of functions defined by automatons with finite memory, conference abstract (2013) (Ukrainian). Available from: https://www.researchgate.net/publication/311414454

S. Serbenyuk, On two functions with complicated local structure, conference abstract (2013). Available from: https://www.researchgate.net/publication/311414256

Symon Serbenyuk, Representation of real numbers by the alternating Cantor series, slides of talk (2013) (Ukrainian). Available from: https://www.researchgate.net/publication/303720347

Symon Serbenyuk, Representation of real numbers by the alternating Cantor series, preprint (2013) (Ukrainian). Available from: https://www.researchgate.net/publication/316787375

Symon Serbenyuk, Defining by functional equations systems of one class of functions, whose argument defined by the Cantor series, conference talk (2014) (Ukrainian). Available from: https://www.researchgate.net/publication/314426236

Symon Serbenyuk, Applications of positive and alternating Cantor series, slides of talk (2014) (Ukrainian). Available from: https://www.researchgate.net/publication/303736670

S. O. Serbenyuk, Defining by functional equations systems of one class a functions, whose arguments defined by the Cantor series, conference abstract (2014) (Ukrainian). Available from: https://www.researchgate.net/publication/311415359

S. O. Serbenyuk, Functions, that defined by functional equations systems in terms of Cantor series representation of numbers, Naukovi Zapysky NaUKMA 165 (2015), 34–40 (Ukrainian). Available from: https://www.researchgate.net/publication/292606546

S.O. Serbenyuk, Nega-Q̃-representation of real numbers, conference abstract (2015). Available from: https://www.researchgate.net/publication/311415381

S.O. Serbenyuk, On one function, that defined in terms of the nega-Q̃-representation, from a class of functions with complicated local structure, conference abstract (2015) (Ukrainian). Available from: https://www.researchgate.net/publication/311738798

S. Serbenyuk, Nega-Q̃-representation as a generalization of certain alternating representations of real numbers, Bull. Taras Shevchenko Natl. Univ. Kyiv Math. Mech. 1 (35) (2016), 32–39 (Ukrainian). Available from: https://www.researchgate.net/publication/308273000

S.O. Serbenyuk, On one class of functions that are solutions of infinite systems of functional equations, preprint (2016), arXiv: 1602.00493

S. Serbenyuk, On one class of functions with complicated local structure, Šiauliai Mathematical Seminar 11 (19) (2016), 75–88.

Symon Serbenyuk, On one nearly everywhere continuous and nowhere differentiable function that defined by automaton with finite memory, preprint (2017), arXiv: 1703.02820

S.O. Serbenyuk, Continuous functions with complicated local structure defined in terms of alternating Cantor series representation of numbers, Zh. Mat. Fiz. Anal. Geom. 13 (2017), 57–81. https://doi.org/10.15407/mag13.01.057

S. Serbenyuk, Representation of real numbers by the alternating Cantor series, Integers 17 (2017), Paper No. A15, 27 pp.

K. Weierstrass, Über continuierliche Functionen eines reellen Argumentes, die für keinen Werth des letzeren einen bestimmten Differentialquotienten besitzen, Math. Werke 2 (1895), 71–74 (German).

W. Wunderlich, Eine überall stetige und nirgends differenzierbare Funktion, Elemente der Math. 7 (1952), 73–79 (German).

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Опубліковано

2018-07-11

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(1)
Serbenyuk, S. Non-Differentiable Functions Defined in Terms of Classical Representations of Real Numbers. J. Math. Phys. Anal. Geom. 2018, 14, 197-213.

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