A New Estimate on the Rate of Convergence of Durrmeyer-Bézier Operators

Abstract

We obtain an estimate on the rate of convergence of Durrmeyer-Bézier operaters for functions of bounded variation by means of some probabilistic methods and inequality techniques. Our estimate improves the result of Zeng and Chen (2000).

1. Introdution

In 2000, Zeng and Chen [1] introduced the Durrmeyer-Bézier operators which are defined as follows:

(1.1)

where is defined on , , , , are Bézier basis functions, and , are Bernstein basis functions.

When , is just the well-known Durrmeyer operator

(1.2)

Concerning the approximation properties of operators and some results on approximation of functions of bounded variation by positive linear operators, one can refer to [2–7]. Authors of [1] studied the rate of convergence of the operators for functions of bounded variation and presented the following important result.

Theorem A

Let be a function of bounded variation on , (), then for every and one has

(1.3)

where is the total variation of on and

(1.4)

Since the Durrmeyer-Bézier operators are an important approximation operator of new type, the purpose of this paper is to continue studying the approximation properties of the operators for functions of bounded variation, and give a better estimate than that of Theorem A by means of some probabilistic methods and inequality techniques. The result of this paper is as follows.

Theorem 1.1.

Let be a function of bounded variation on , (), then for every and one has

(1.5)

where is defined in (1.4).

It is obvious that the estimate (1.5) is better than the estimate (1.3). More important, the estimate (1.5) is true for all . This is an important improvement comparing with the fact that estimate (1.3) holds only for .

2. Some Lemmas

In order to prove Theorem 1.1, we need the following preliminary results.

Lemma 2.1.

Let be a sequence of independent and identically distributed random variables, is a random variable with two-point distribution (and is a parameter). Set with the mathematical expectation and with the variance Then for one has

(2.1)

(2.2)

Proof.

Since , from the distribution series of, by convolution computation we get

(2.3)

Furthermore by direct computations we have

(2.4)

Thus we deduce that

(2.5)

By Schwarz's inequality, it follows that

(2.6)

The inequality (2.1) is proved.

Similarly, by using the identities

(2.7)

we get the inequality (2.2). Lemma 2.1 is proved.

Lemma 2.2.

Let , be Bernstein basis functions, and let be Bézier basis functions, then one has

(2.8)

Proof.

Note that , and Thus

(2.9)

Now by inequality (2.1) of Lemma 2.1 we obtain

(2.10)

Similarly, by using inequality (2.2), we obtain

(2.11)

Thus Lemma 2.2 is proved.

3. Proof of Theorem 1.1

Let satisfy the conditions of Theorem 1.1, then can be decomposed as

(3.1)

where

(3.2)

Obviously thus from (3.1) we get

(3.3)

We first estimate , from [1, page 11] we have the following equation:

(3.4)

where .

Thus by Lemma 2.2, we get . Note that , we have

(3.5)

Next we estimate . From (15) of [1], it follows the inequality

(3.6)

That is,

(3.7)

On the other hand, note that , we have

(3.8)

From (3.7) and (3.8) we obtain

(3.9)

Using inequality

(3.10)

we get

(3.11)

Thus from (3.9) we obtain

(3.12)

Theorem 1.1 now follows by collecting the estimations (3.3), (3.5), and (3.12).