Localized and Local Fractional Derivative of the Takagi Function
DOI:
https://doi.org/10.52575/2687-0959-2025-57-2-93-110Keywords:
Localized Fractional Derivative, Local Fractional Derivative, Takagi FunctionAbstract
In this paper it is proved that localized fractional derivatives of Riemann – Liouville type of order 0 < α < 1 are bounded from the Hölder space with exponent λ, 0 < λ ≤ 1 and logarithmic factor into the Hölder space with exponent λ − α, 0 < λ − α and logarithmic factor. Localized and local fractional derivatives, minimum and maximum points of the Takagi function are calculated. It is shown that the Takagi function belongs to the Hölder space with exponent one and logarithmic factor.
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References
Grinko AP. Localized derivatives in spaces of functions representable by localized fractional integrals. Integral Transforms and Special Functions. 2019;30(10):817-832.
Grinko AP. Generalized Abel type integral equations with localized fractional integrals and derivatives. Integral Transforms and Special Functions. 2018;29(6):489-504.
Grin’ko AP. Localized and local derivatives of fractional order of functions with a given modulus of continuity. Applied Mathematics & Physics. 2024;56(4):296-313.
Takagi T. A simple example of a continuous function without derivative. Proc. Phys. Math. Soc. Japan. 1903;1:176-177.
Hata M., Yamaguti M. Takagi function and its generalization Japan J. Appl. Math. 1984;1:183-199.
Samko SG., Kilbas AA., Marichev OI. Integrals and derivatives of fractional order and some of their applications. Mn.; Science and Technology; 1987. 687 p.
Shidfar A., Sabetfakhri K. On the Continuity of Van Der Waerden’s Function in the Hölder Sense. Amer. Math. Monthly. 1986;93(5):375-376.
Sheipak IA., On H¥older exponents of self-similar functions Functional Analysis and Its Applications 2019;53(1):67–78.
Grinko AP. Localized fractional derivative of Djrbashian-Caputo type. Integral Transforms and Special Functions. 2021;32(12);1002-1018.
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Muskhelishvili NI. Singular integral equations. - M.: Nauka; 1968. 511 p. (In Russ.)
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