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.