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A new kind of Bernstein-Schurer-Stancu-Kantorovich-type operators based on \(q\)-integers. (English) Zbl 1359.41010

Summary: P. N. Agrawal et al. [Boll. Unione Mat. Ital. 8, No. 3, 169–180 (2015; Zbl 1331.41026)] introduced a Stancu-type Kantorovich modification of the operators proposed by M.-Y. Ren and X.-M. Zeng [Bull. Korean Math. Soc. 50, No. 4, 1145–1156 (2013; Zbl 1276.41005)] and studied a basic convergence theorem by using the Bohman-Korovokin criterion, the rate of convergence involving the modulus of continuity, and the Lipschitz function. The concern of this paper is to obtain Voronovskaja-type asymptotic result by calculating an estimate of fourth order central moment for these operators and discuss the rate of convergence for the bivariate case by using the complete and partial moduli of continuity and the degree of approximation by means of a Lipschitz-type function and the Peetre \(K\)-functional. Also, we consider the associated GBS (generalized Boolean sum) operators and estimate the rate of convergence for these operators with the help of a mixed modulus of smoothness. Furthermore, we show the rate of convergence of these operators (univariate case) to certain functions with the help of the illustrations using Maple algorithms and in the bivariate case, the rate of convergence of these operators is compared with the associated GBS operators by illustrative graphics.

MSC:

41A35 Approximation by operators (in particular, by integral operators)
41A25 Rate of convergence, degree of approximation
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
41A28 Simultaneous approximation

Software:

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References:

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