Kantorovich Variant of α-Bernstein Operators using Contagion Distribution

This study puts forth an enhancement to the α-Bernstein operators, with the help of the Pólya-Eggenberger distribution, also known as the contagion distribution for m ≥ 2. In order to deal with integrable functions we utilize the parameter γto be of the order of 1/m, to define and introduce the Kantorovich variation of the above said operators. First, we give some auxiliary properties and then we give an upper bound of our proposed operators on the space of Lebesgue integrable functions, L1[0, 1], and the space of continuous functions, C[0, 1]. Next, we study its asymptotic results with the help of Taylor’s expansion. Modulus of continuity is also used to provide the asymptotic properties of Kantorovich operators, both for Lebesgue and continuous spaces of functions. Moreover, like the classical Kantorovich-Bernstein operators, we will see that the Kantorovich variant defined in this paper also does not preserve e1, that is, K〈γ〉m,α (t; u)≠u.  Rather, we get an expression which tends to u as m tends to ∞. As an opening at finding a better approximating linear operators, we try and preserve these operators at e1 and propose a genuine-type modification for same. We have also included graphical illustrations to help analyze and compare the approximation results and properties of both the Kantorovich variant of the α-Bernstein operators using contagion distribution and its genuine-type modification.