Progress in Fractional Differentiation & Applications

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Here we consider very general stochastic positive linear operators induced by general positive linear operators that are acting on continuous functions in the trigonometric sense. These are acting on the space of real conformable fractionally differentiable stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related trigonometric conformable fractional stochastic Shisha-Mond type inequalities of Lq-type 1 ≤ q < ∞ and corresponding trigonometric conformable fractional stochastic Korovkin type theorems. These are regarding the trigonometric stochastic q-mean conformable fractional convergence of a sequence of stochastic positive linear operators to the stochastic unit operator for various cases. All convergences are derived with rates and are given via the trigonometric conformable fractional stochastic inequalities involving the stochastic modulus of continuity of the α-th conformable fractional derivatives of the engaged stochastic process, α ∈ (n,n+1), n ∈ Z+. The impressive fact is that only two basic real Korovkin test functions assumptions, one of them trigonometric, are enough for the conclusions of our trigonometric conformable fractional stochastic Korovkin theorems. We give applications to stochastic Bernstein operators in the trigonometric sense.

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