A Sequence of Kantorovich-Type Operators on Mobile Intervals

Abstract

In this paper, we introduce and study a new sequence of positive linear operators, acting on both spaces of continuous functions as well as spaces of integrable functions on [0, 1]. We state some qualitative properties of this sequence and we prove that it is an approximation process both in C([0, 1]) and in L p ([0, 1]), also providing some estimates of the rate of convergence. Moreover, we determine an asymptotic formula and, as an application, we prove that certain iterates of the operators converge, both in C([0, 1]) and, in some cases, in L p ([0, 1]), to a limit semigroup. Finally, we show that our operators, under suitable hypotheses, perform better than other existing ones in the literature.

Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

“…where $$$ \left({}_{0+}{I}_1^{\beta }f\right)(t)={\int}_0^1\frac{{\left(1-t\right)}^{\beta -1}}{\Gamma \left(\beta \right)}f(t) dt $$$, for more information about Bernstein–Kantorovich operators, Grüss–Voronovskaya type theorems, and fractional type operators, readers can refer to the previous works [5–26]. Additionally, we introduce key definitions employed within the context of this paper, such as fractional integral of Riemann–Liouville [27] and the modulus of continuity [28].…”