Kantorovich‐type generalization of parametric Baskakov operators

Abstract

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.

Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

“…For example, Mohiuddine et al [26] introduced the family of α-Bernstein-Kantorovich operators and associated bivariate form and demonstrated the results regarding the rate of convergence via Peetre's K -functional together with modulus of continuity. In addition to this, the Stancu type α-Bernstein-Kantorovich, α-Baskakov and their Kantorovich form, and α-Baskakov-Durrmeyer operators were analyzed by Mohiuddine and Özger [32], Aral et al [6,20], and Mohiuddine et al [31], respectively, and for other blending type operators, see [23,27,36]. Some other modifications of Bernstein operators have been studied in [2,15,16,25,33,34,37,42].…”