The present paper deals with the bivariate {(p,q)}-Baskakov–Kantorovich operators and their approximation properties. First we construct the operators and obtain some auxiliary results such as calculations of moments and central moments, etc. Our main results consist of uniform convergence of the operators via the Korovkin theorem and rate of convergence in terms of modulus of continuity.
In this manuscript, we define a Kantorovich generalization of the nonnegative parametric Baskakov operators. After that, the weighted uniform convergence of the generalized operators is proved. Also, we present Voronovskaja-type asymptotic approximation theorem then establish weighted approximation properties for parametric Kantorovich operator. Numerical results show that depending on the value of the parameter, we obtain a better approximation. KEYWORDS-Kantorovich operator, Voronovskaja-type theorem, weighted approximation MSC CLASSIFICATION 41A36; 41A25; 26A15 INTRODUCTIONThe primary focus of the approximation theory is to approximate real-valued continuous functions by a simpler class of functions such as algebraic polynomials. Such topics have attracted the attention of many mathematicians. Kantorovich 1 presented a class of positive linear operators on bounded the interval [0, 1] for Lebesgue integrable functions defined on the interval [0, 1]. This class of operators has been studied by many researchers. Butzer 2 studied Voronovskaja-type results for the Kantorovich polynomials. Abel 3 gave an approximation as well as the convergence rate of these polynomials. On the other hand, Baskakov 4 introduced a sequence of positive linear operators, called Baskakov operators, on unbounded the interval [0, ∞) for suitable functions defined on the interval [0, ∞). Later, the Baskakov operators have been studied by many researchers. Pethe 5 studied approximation properties of Baskakov operators. Gupta 6 studied the rate of convergence of Baskakov-type operators. Mihesan 7 constructed the generalization of the Baskakov operators and the convergence rate of the generalization obtained by Wafi and Khatoon. 8 Moreover, the preservation properties of the Baskakov-Kantorovich operators are studied by Chungou and Zhihui. 9 A novel collection of similar studies can be found in the book by Gupta and Tachev. 10 On the other hand, q-analogous to the Baskakov operators were introduced by Aral and Gupta. 11 The same authors introduced another q-analogous to the Baskakov operators and studied the convergence rate in weighted norm and some shape preserving properties. 12 Recently, Aral and Erbay 13 proposed -Baskakov operators and studied their approximation properties. They showed that even though convergence is independent of the parameter , the approximation errors depend on .The aim of the manuscript is to define a Kantorovich generalization of the nonnegative parametric Baskakov operators introduced by Aral and Erbay. 13 We name it -Baskakov-Kantorovich operator. Later, we calculate the moments and central moments of the new operator. The weighted uniform convergence of the generalized operators is proved.
Abstract. This work relates to bivariate Bernstein-Chlodowsky operator which is not a tensor product construction. We show that the operator preserves some properties of the original function, for example; function of modulus of continuity, Lipschitz constant, and a kind of monotony.Mathematics subject classification (2010): 41A25, 41A35, 41A36.
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