Certain Functions Defined in Terms of Cantor Series
DOI:
https://doi.org/10.15407/mag16.02.174Анотація
Цю статтю присвячено деяким прикладам функцій, аргумент яких подано в термінах рядів Кантора.Mathematics Subject Classification: 26A27, 26A30, 11B34, 11K55
Ключові слова:
ніде недиференційовна функція, сингулярна функція, розвинення дійсного числа, немонотонна функція, розмірність ГаусдорфаПосилання
K.A. Bush, Continuous functions without derivatives, Amer. Math. Monthly 59 (1952), No. 4, 222–225. https://doi.org/10.1080/00029890.1952.11988110
G. Cantor, Ueber die einfachen Zahlensysteme, Z. Math. Phys. 14 (1869), 121–128 (German).
G.H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), 301–325. https://doi.org/10.2307/1989005
J. Hančl, R. Tijdeman, On the irrationality of factorial series, Acta Arith. 118 (2005), No. 4, 383–401. https://doi.org/10.4064/aa118-4-5
J. Gerver, More on the differentiability of the Rieman function, Amer. J. Math. 93 (1971), 33–41. https://doi.org/10.2307/2373445
H. Minkowski, Zur Geometrie der Zahlen. In: H. Minkowski (ed.) Gesammeine Abhandlungen, 2, Druck und Verlag von B.G. Teubner, Leipzig und Berlin, 1911, 50–51 (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. Serbenyuk, On one class of functions with complicated local structure, Šiauliai Math. Semin. 11 (19) (2016), 75–88.
S.O. Serbenyuk, Functions 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, 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
Symon Serbenyuk, On one application of infinite systems of functional equations in function theory, Tatra Mt. Math. Publ. 74 (2019), 117–144. https://doi.org/10.2478/tmmp-2019-0024
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. Serbenyuk, Representation of real numbers by the alternating Cantor series, Integers 17 (2017), Paper No. A15.
S. Serbenyuk, On one fractal property of the Minkowski function, Revista de la Real Academia de Ciencias Exactas, Fı́sicas y Naturales. Serie A. Matemáticas 112 (2018), No. 2, 555–559. https://doi.org/10.1007/s13398-017-0396-5
S.O. Serbenyuk, Non-differentiable functions defined in terms of classical representations of real numbers, Zh. Mat. Fiz. Anal. Geom. 14 (2018), 197–213. https://doi.org/10.15407/mag14.02.197
S. Serbenyuk, One one class of fractal sets, preprint, https://arxiv.org/abs/ 1703.05262.
S. Serbenyuk, More on one class of fractals, preprint, https://arxiv.org/abs/ 1706.01546.
S.O. Serbenyuk, One distribution function on the Moran sets, Azerb. J. Math. 10 (2020), No. 2, 12–30.
Liu Wen, A nowhere differentiable continuous function constructed using Cantor series, Math. Mag. 74 (2001), No. 5, 400–402. https://doi.org/10.2307/2691039
W. Wunderlich, Eine überall stetige und nirgends differenzierbare Funktion, El. Math. 7 (1952), 73–79 (German).