Generalization of the Tribin Function

We consider the Tribin function and its generalization based on the \( {Q}_s^{\ast } \) -representation of real numbers, which is an s-symbol encoding of numbers and, generally speaking, a nonself-similar generalization of the s-adic representation. By definition, the function f associates the number \( x={\varDelta}_{\upalpha_1{\upalpha}_2\dots {\upalpha}_n\dots}^{Q_s^{\ast }}, \) where αn ∈ L ≡ As × As × … × As × … and As = {0, 1, …, s − 1} is an alphabet, s ≥ 3; with the number \( y=f(x)={\varDelta}_{\upgamma_1{\upgamma}_2\dots {\upgamma}_n\dots}^{G_2^{\ast }}, \)

$$ {\gamma}_1=\left\{\begin{array}{l}0\kern1em \mathrm{for}\kern1em {\upalpha}_1=0,\\ {}1\kern1em \mathrm{for}\kern1em {\upalpha}_1\ne 0,\end{array}\right.{\gamma}_{n+1}=\left\{\begin{array}{l}{\gamma}_n\kern2.5em \mathrm{for}\kern1em {\upalpha}_{n+1}={\upalpha}_n,\\ {}1-{\gamma}_n\kern1em \mathrm{for}\kern1em {\upalpha}_{n+1}\ne {\upalpha}_n,\end{array}\right. $$

where the \( {G}_2^{\ast } \) -representation of numbers has the two-symbol alphabet A2 = {0, 1}. We prove that the function f is well-defined, continuous, and nowhere monotone. Its variational properties are also analyzed.

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Correspondence to O. M. Baranovs’kyi.

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Translated from Neliniini Kolyvannya, Vol. 22, No. 3, pp. 380–390, July–September, 2019.

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Prats’ovytyi, M.V., Baranovs’kyi, O.M. & Maslova, Y.P. Generalization of the Tribin Function. J Math Sci 253, 276–288 (2021). https://doi.org/10.1007/s10958-021-05227-3

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