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Here we consider the quantitative Caputo and Canavati fractional approximation of positive sublinear operators to the unit operator. At the beginning we perform the investigation of the fractional rate of the convergence of the Bernstein-Kantorovich-Choquet and Bernstein-Durrweyer-Choquet polynomial Choquet-integral operators. After that we discuss the very general comonotonic positive sublinear operators based on the representation theorem of Schmeidler (1986) [1]. We ended with the approximation by the very general direct Choquet-integral form positive sublinear operators. All fractional approximations are presented via inequalities implying the modulus of continuity of the approximated function fractional order derivative.

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