On Chlodowsky Variant of Baskakov Type Operators

Atakut and Ibrahim Büyükyaz?c?

Ankara University,Faculty of Science,Department of Mathematics,06100 Tandogan,Ankara,Turkey

Abstract.In the present paper,we deal with Chlodowsky type generalization of the Baskakov operators,special case of these operators includes Chlodowsky type Meyer–K?nig and Zeller operators(see[21]). With the help of Bohman-Korovkin theorem,we obtain some approximation properties for these operators.We give a modification of the operators in the space of differentiable functions and we also present examples of graphs for approximation.Finally,we apply these operators to the solution of the differential equation.

Key Words:Approximation properties,Rate of convergence,Chlodowsky type MKZ operators,Baskakov operators,Differential equation.

1 Introduction and Construction of the Operators

Let{?n}(n=1,2,···),be a sequence of functions,having the following properties:

i)?nis analytic on a domain D containing the diskfor each positive integer n,

ii)?n(0)=1 for n∈N,

iii)?n(x)>0 andfor every positive integer n,x ∈[0,∞)and and for every nonnegative integer k.

In[5],V.A.Baskakov introduced following the sequence of linear operators{Ln},

Furthermore,if we take ?n(x)=(1+x)nand replace x by anx in the operator(1.1),we have

This operator is known as Bernstein type rational function which was studied in[4,16].

where for 0

Motivated by this work,we give Chlodowsky type generalization of Ln(f;x)operators given by(1.2)as follows:

and γ be real number in the intervalAssume that the sequence of functions{?n}satisfies the conditions(i)-(iv).For a function f defined on[0,∞)and bounded on every finite interval[0,γ],we define the following sequence of linear positive operators:

Recently,linear positive operators and their Chlodowsky type generalizations have been widely studied by several authors[1,3–22],because this generalization allows us to investigate approximation properties of functions defined on the infinite interval[0,∞)by using the similar techniques and methods on the classical operators.

The aim of this paper is to study some convergence properties of the operators Ln(f;βn,x)defined by(1.4)and modify the operators for differentiable functions,in order to improve the rate of convergence on the interval[0,βn]extending infinity as n→∞.Also we give an application to functional differential equation by using these operators.

2 Approximation properties of Ln(f;βn,x)

In this section we study conditions of Korovkin theorem[2]and the rate of convergence,an asymptotic formula for the operators(1.4)for f ∈C[0,γ].

Now we use the test functions ei(t)=ti,i=0,1,2.Then,we obtain the following result.

Theorem 2.1.Let{βn}be a positive increasing sequence satisfying(1.3)and the operators Ln(f;βn,x)be defined by(1.4)with the conditions(i)-(iv).For every finite intervaland for eachwe have

Proof.Firstly,from condition(i),we have

so we get

By the definition of the operators(1.4)and using e1(t)=t

by(iv),we have

or

On the other hand,observe that

combining(2.2)with(2.3),we have

and hence,

uniformly in[0,γ].

Finally,for e2(t)=t2,we obtain

by(iv),we get

using this equalities,we obtain

which implies that

on the other hand,it is clear that Ln(e2(t);βn,x)?e2(x)≥0.Using(2.4),we get

Hence we have

uniformly in[0,γ].Thus,the proof of the theorem is completed.

Remark 2.1.LetSimple calculations,one can easily obtain ψn=1,αn,k=0.Then Ln(f;βn,x)has the following form:

Chlodowsky type MKZ operators(see[21]).

Example 2.1.For n=10,50 andthe convergence of Ln(f;βn,x)to f(x)will be illustrated in Fig.1 and Fig.2.

Example 2.2.Let n=10.For βn=n4/5and βn=n2/3,the convergence of Ln(f;βn,x)to f(x)will be illustrated in Fig.3.

Figure 1:The convergence of Chlodowsky type MKZ operators to f(x)=x1/3.

Figure 2:The convergence of Chlodowsky type MKZ operators to

Figure 3:The convergence of Chlodowsky type MKZ operators to

Now we give the approximation order of operators(1.4)with help of asymptotic inequality.

Let I=[0,γ]and C(I)be the space of all continuous functions f.For a fixed r∈N we denote by

Theorem 2.2.If the operators Lnare defined by(1.4),then for sufficiently large n and for every f ∈C2(I)

Proof.By the Taylor formula,we write

where λ(t)→0 as t→x.If we Apply the operators(1.4)to(2.5),we get

From(2.1),(2.2)and(2.4),we have

using(2.7a)-(2.7c)in(2.6),one obtains

and hence

which implies

Thus the desired result is obtained.

3 A generalization of the Ln(f;βn,x)

In recent years several authors[7,8,17,21,22]investigated approximation properties of certain linear operators for differentiable functions.In this section we will modify the operator(1.4)for differentiable functions,in order to improve the rate of convergence of the sequence{Lnf}to f(see Example 3.1)

Theorem 3.1.Ifthen

where B(α,r)is a beta function and Lnis defined by(1.4).

Proof.From(2.1),we can write

Consider the term in parentheses.Using modified Taylor’s formula we have

since f(r)∈LipMα,we obtain

On the other hand,we have

where B(α,r)is a beta function.Therefore we write

from(3.1)and(3.2),we have

Thus the proof of theorem is completed.

Remark 3.1.If we choosewe obtain Chlodowsky type generalized MKZ operators defined by

(see[21]).

Example 3.1.For n=10,βn=n2/3and r=2,the convergence comparison ofand Ln(f;βn,x)to f(x)will be illustrated in Fig.4 and Fig.5.

Example 3.2.Let n=10.For βn=n4/5and βn=n2/3,the convergence ofto f(x)=x6will be illustrated in Fig.6

4 Application an differential equation

Many authors obtained some differential equations by using the linear positive operators which are solution of these equations,we refer the readers to[1,3,8,9,19].In this section,using the same idea and method,as an application to the differential equation,we obtain a functional differential equation so that the linear positive operator Ln(f;βn,x)is a particular solution of it.

Figure 4:Comparison Chlodowsky type MKZ operators and Chlodowsky type MKZ-Taylor operators for f(x)=

Figure 5:Comparison Chlodowsky type MKZ operators and Chlodowsky type MKZ-Taylor operators for f(x)=

Figure 6:The convergence of Chlodowsky type MKZ–Taylor operators to f(x)=x6.

Theorem 4.1.LetFor each x ∈[0,γ]and f ∈C[0,γ],the operators Ln(f;βn,x)defined by(1.4)satisfy the following differential equation:

where

Proof.By the Theorem 2.1,if f ∈C(I),then Ln(f;βn,x)converges uniformly to f(x)on[0,γ].So we can differentiate both sides of(1.4)term by term to obtain

Thus we can see that:

and

hence we get

This gives the desired result.

Corollary 4.1.The operators Ln(f;βn,x)given by(1.4)are a particular solution of the following differential equation:

where βnis given by(4.2).

Proof.Selecting f(t)=βn?t in(4.1),we get

from(2.1),we have

which gives the proof.