Abstract
We give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke‐type polynomials and obtain convergence properties of these operators by using Korovkin′s theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre′s K‐functional. Furthermore, an example of Kantorovich type of the operators including Gould‐Hopper polynomials is presented and Voronovskaya‐type result is given for these operators including Gould‐Hopper polynomials.
Highlights
The Szasz operators called Szasz-Mirakyan operators which are defined by 1 Sn f ; x : ∞ e−nx k0 nx k! k f k n1.1 where n ∈ N, x ≥ 0, and f ∈ C 0, ∞ have an important role in the approximation theory, and their approximation properties have been investigated by many researchers.In 2, Jakimovski and Leviatan proposed a generalization of Szasz operators by means of the Appell polynomials pk x which have the generating functions of the form:g t etx pk x tk, k0Abstract and Applied Analysis where g z∞ k ak zk a0 / 0 is an analytic function in the disc |z| < R, R>1 and g 1
Varma et al 4 defined another generalization of Szasz operators by means of the Brenke-type polynomials
In the case of B t et and A t 1, with the help of 1.7, it follows that pk x xk/k!, so the operators 1.11 reduce to the Szasz-Mirakyan-Kantorovich operators defined by 6
Summary
Under the assumption that pk x ≥ 0 for x ∈ 0, ∞ , Jakimovski and Leviatan 2 , defined the following linear positive operators: Pn f ; x : e−nx g1. With the help of these polynomials, Ismail constructed the following linear positive operators: Tn f ; x : e−nxH 1 A1. Varma et al 4 defined another generalization of Szasz operators by means of the Brenke-type polynomials. The aim of this paper is to present a Kantorovich type of the operators given by 1.10 and to give their some approximation properties. In the case of B t et and A t 1, with the help of 1.7 , it follows that pk x xk/k!, so the operators 1.11 reduce to the Szasz-Mirakyan-Kantorovich operators defined by 6. As an example, we present a Kantorovich type of the operators including Gould-Hopper polynomials and we give a Voronovskaya-type theorem for the operators including Gould-Hopper polynomials
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