Abstract
In this paper we introduce q-Szasz-Mirakjan-Kantorovich operators generated by a Dunkl generalization of the exponential function and we propose two different modifications of the q-Szasz-Mirakjan-Kantorovich operators which preserve some test functions. We obtain some approximation results with the help of the well-known Korovkin theorem and the weighted Korovkin theorem for these operators. Furthermore, we study convergence properties in terms of the modulus of continuity and the class of Lipschitz functions. This type of operator modification enables better error estimation than the classical ones. We also obtain a Voronovskaja-type theorem for these operators.
Highlights
Introduction and preliminariesIn, Bernstein [ ] introduced a sequence of operators Bn : C[, ] → C[, ] defined by n Bn(f, x) =n xk( – x)n–kf k, x ∈ [, ], k n ( . ) k=for n ∈ N and f ∈ C[, ]
We have show that our modified operators have a better error estimation than the classical ones
We prove the Voronovskaja-type result for our modified Dunkl analog of q-SzászMirakjan-Kantorovich operators Kn,qn
Summary
Içöz gave a Dunkl generalization of Szász-Mirakjan-Kantorovich operators in [ ] and one gave a. Let un,q(x) be the following sequence of real valued continuous function defined on [ , ∞) with ≤ un,q(x) < ∞: un,q(x). N. we consider the following linear positive operators: Kn∗,q(f ; x) = Kn,q f ; un,q(x). Proof The proof is based on the well known Korovkin’s theorem regarding the convergence of a sequence of linear and positive operators; so, it is enough to prove the conditions lim n→∞. I.e., the rate of convergence of the operators Kn,qn by means of an element of the Lipschitz class functions is better than the ordinary error estimation given by From ( . ), it follows that the above claim holds for Theorem . , i.e., the rate of convergence of the operators Kn,qn by means of an element of the Lipschitz class functions is better than the ordinary error estimation given by ( . ), where
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