On the rate of convergence of Bernstein polynomials of functions of bounded variation

On the Rate of Convergence of Two Bernstein–Bézier Type Operators for Bounded Variation Functions

“…The proof of this lemma is based on the method of Bojanic and Vuillemier [1] (see also Cheng [4]). We decompose [0, 1] into three parts:…”

“…In 1983 Cheng [4] gave a rate of convergence of B n for normalized bounded variation functions as follows:…”

On the Rate of Convergence of the Generalized Durrmeyer Type Operators for Functions of Bounded Variation

“…First, with the method of Bojanic and Vuilleumier [1] (see Cheng [4] and Guo [6]), we prove the following: Lemma 1. For every x # (0, 1) and n>1Âx(1&x), there holds…”

“…Cheng [4,5] and Guo [6] estimated the rates of convergence of Bernstein operators B n ( f, x), Sza sz operators S n ( f, x) and Durrmeyer operators D n ( f, x) for functions of bounded variation and proved that the operators B n ( f, x), S n ( f, x) and D n ( f, x) all convergence to the limit 1 2 f (x+ )+ 1 2 f (x& ) for functions of bounded variation. References [2,3] proved that the operators B n, : ( f, x), L n, : ( f, x) and S n, : ( f, x) all convergence to the limit (1Â2 : ) f (x+ )+(1&1Â2 : ) f (x& ) for functions of bounded variation (: 1, n Ä + ).…”

Rate of convergence of Chlodowsky type Durrmeyer Jakimovski-Leviatan operators

“…For example, Bojanic and Vuilleumier [1] estimated the rate of convergence of Fourier Legendre series of functions of bounded variation on the interval [0, 1], Cheng F. [3] estimated the rate of convergence of Bernstein polynomials of functions of bounded variation on the interval [0, 1], Zeng and Chen [16] estimated the rate of convergence of Durrmeyer type operators for functions of bounded variation on the interval [0,1]. On the other hand, in recent years, there is an increasing interest in modifying linear operators so that the new versions reproduce some basic functions, e.g.…”