Approximation Properties of Generalized λ-Bernstein–Stancu- Type Operators

Fujian Provincial Key Laboratory of Data-Intensive Computing, Key Laboratory of Intelligent Computing and Information Processing, School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China Kastamonu University, Faculty of Education, Mathematics and Science Education, Kastamonu, Turkey Gazi University, Faculty of Science, Department of Mathematics, Ankara, Turkey


Lemma 1. For generalized λ-Bernstein-Stancu operators
with shifted knots, we have the following equalities: Journal of Mathematics

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Proof. If we use Bézier basis functions (8) in λ-Bernstein-Stancu operators (7), we obtain where anks to the linearity of Bernstein-Stancu operators (3), we obtain Now, we will compute Υ α,β  (s, x), we obtain the following equation: en, we have the following equality for the third moment by using the linearity of G α,β m,λ (g(s), x): where Υ α,β Journal of Mathematics Combining (17) Using Lemma 1 and the linearity of G α,β m,λ (g(s), x), we have the following Corollary 2.

Corollary 2.
We obtain the following equalities:  14 Journal of Mathematics   and we obtain

Convergence Properties of
Using Cauchy-Schwarz inequality, we obtain us, we proved eorem 4.

Numerical Examples
In this section, we show the theoretical results demonstrated in the previous sections by the following example.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

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