Abstract
In this work, we extend the works of F. Usta and construct new modifiedq-Bernstein operators using the second central moment of theq-Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’sK-functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.
Highlights
In [1], Phillips introduced q-analogue of Bernstein operators as follows: ! lBql ðζ ; zÞ = 〠 ζ i=01⁄2iq 1⁄2lq pql,iðzÞ, z ∈ 1⁄20, 1, ð1Þ where pql,iðzÞ =! l zið1 − zÞlq−i, i = 0, 1, ⋯, l, and ζ ∈ C1⁄20, 1. i qLater, generalizations of q-Bernstein operators (1) attracted a lot of interest and were constructed and researched widely by a number of researchers
In [2], Mahmudov and Sabancigil introduced q-Bernstein-Kantorovich operators and studied local and global approximation properties
[3], Acu et al defined modified q-Bernstein-Kantorovich operators and established the shape-preserving properties of these operators, e.g., monotonicity and convexity
Summary
In [1], Phillips introduced q-analogue of Bernstein operators as follows:. Later, generalizations of q-Bernstein operators (1) attracted a lot of interest and were constructed and researched widely by a number of researchers. There are many papers about the research and application of q-operators, and we mention some of them: q-BleimannButzer-Hahn operators [15], Bivariater q-Meyer-König-Zeller operators [16], q-Baskakov operators [17, 18], q-MeyerKönig-Zeller-Durrmeyer operators [19], q-Phillips operators [20, 21], q-Szász operators [22], q-Bernstein operators [23], and so on. All this achievement motivates us to construct the q-analogue of the operators (2).
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