Abstract
In this paper, we introduce a family of bivariate α , q -Bernstein–Kantorovich operators and a family of G B S (Generalized Boolean Sum) operators of bivariate α , q -Bernstein–Kantorovich type. For the former, we obtain the estimate of moments and central moments, investigate the degree of approximation for these bivariate operators in terms of the partial moduli of continuity and Peetre’s K-functional. For the latter, we estimate the rate of convergence of these G B S operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness.
Highlights
Since the famous Bernstein polynomial was proposed in 1912, the study of Bernstein type operators has not ceased
We introduce the bivariate α, q-Bernstein–Kantorovich operators as follows: for f ∈ C ( I 2 ), I = [0, 1] × [0, 1], 0 < q1, q2 < 1 and α1, α2 are any fixed real numbers in [0, 1], (α,α )
In order to prove the main conclusion of this paper, the following lemmas are given: Lemma 1. (See [3]) The following equalities hold: Tn,q,α (1; x ) = 1, Tn,q,α (t; x ) = x, x (1 − x ) (1 − α ) q n −1 [ 2 ] q x (1 − x )
Summary
Since the famous Bernstein polynomial was proposed in 1912, the study of Bernstein type operators has not ceased. In 2017, Chen et al [1] introduced and studied the monotonic, convex properties and some other important properties of a new generalized positive linear Bernstein operators with parameter α which are defined as n. Mohiuddine et al [2] constructed the Kantorovich type of these family of Bernstein operators (1). Motivated by the research above, we will introduce bivariate α ,α α, q-Bernstein–Kantorovich operators Kn11 ,n22 ,q1 ,q2 ( f ) and GBS operators of bivariate α1 ,α2 α, q-Bernstein–Kantorovich type UKn1 ,n2 ,q1 ,q2 ( f ).
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