We construct Stancu-type Bernstein operators based on Bezier bases with shape parameter λ ∈ [ − 1 , 1 ] and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type λ -Bernstein operators and study their approximation behaviors.

Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ

In this work, we construct a new kind of Bernstein–Schurer operators which includes non-negative real parameter 
$$\alpha $$

. We study some shape preserving properties, namely, monotonicity and convexity of the new operators. We obtain global approximation formula in terms of Ditzian–Totik uniform modulus of smoothness of first and second order and calculate the local direct estimate of the rate of convergence with the help of Lipschitz-type function for our operators. The Voronovskaja-type approximation theorems of the new operators are presented. Finally, in the last section, we provide some graphs, MATLAB code and numerical examples in order to illustrate the significance of our newly constructed operators.

Approximation of functions by a new class of generalized Bernstein–Schurer operators

The approximation theory by linear positive operators investigates how the functions can be best approximated by simpler functions. The most famous basic result for convergence of linear positive operators is due to Weierstrass who introduced an important theorem named Weierstrass’ approximation theorem. At last in 1912 Bernstein introduced the most famous algebraic polynomials Bn.f Ix/ in approximation theory in order to give a constructive proof of Weierstrass’ theorem, which are given by

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Blending type approximation by generalized Bernstein-Durrmeyer type operators

Szász-Durrmeyer type operators based on Charlier polynomials

The present article deals with the modified forms of the Baskakov and Szasz basis functions We introduce a Durrmeyer-type operator having the basis functions in summation and integration due to Stancu (1970) and Pǎltǎnea (2008) We obtain some approximation results, which include the Voronovskaja-type asymptotic formula, local approximation, error estimation in terms of the modulus of continuity, and weighted approximation Also, the rate of convergence for functions with derivatives of bounded variation is established Furthermore, the convergence of these operators to certain functions is shown by illustrative graphics using MAPLE algorithms

Baskakov–Szász-type operators based on inverse Pólya–Eggenberger distribution

In this paper, we consider a certain King type operators which includes general families of Sz´ asz-Mirakjan, Baskakov, Post-Widder and Stancu operators. By introducing two parameter family of Lipschitz type space, which provides global approximationfortheabovementionedoperators,weobtaintherateofconvergence of this class. Furthermore, we give local approximation results by using the first and the second modulus of continuity.

/pdf/local-approximation-properties-for-certain-king-type-2e3zsofd3k.pdf

Ali Mehmet Ozarslan

Papers

Local approximation properties for certain King type operators