A NOTE ON INTEGRAL MODIFICATION OF THE MEYER-KÖNIG … · 2019. 8. 1. · OF THE MEYER-KÖNIG AND ZELLER OPERATORS VIJAY GUPTA and NIRAJ KUMAR Received 6 August 2002 Guo (1988) introduced

IJMMS 2003:31, 2003–2009PII. S0161171203208218

http://ijmms.hindawi.com© Hindawi Publishing Corp.

A NOTE ON INTEGRAL MODIFICATIONOF THE MEYER-KÖNIG AND ZELLER

OPERATORS

VIJAY GUPTA and NIRAJ KUMAR

Received 6 August 2002

Guo (1988) introduced the integral modification of Meyer-König and Zeller opera-tors M̂n and studied the rate of convergence for functions of bounded variation.Gupta (1995) gave the sharp estimate for the operators M̂n. Zeng (1998) gave theexact bound and claimed to improve the results of Guo and Gupta, but there isa major mistake in the paper of Zeng. In the present note, we give the correctestimate for the rate of convergence on bounded variation functions.

2000 Mathematics Subject Classification: 41A30, 41A36.

1. Introduction. For a function defined on [0,1], the Meyer-König and Zelleroperators Pn, n∈N, [8] are defined by

Pn(f ,x)=∞∑k=0pn,k(x)f

(k

n+k), x ∈ [0,1], (1.1)

where

pn,k(x)=(n+k−1

k

)xk(1−x)n. (1.2)

The rates of convergence of some integral modifications of operators (1.1)

were discussed in [2, 4, 5, 6, 7, 10]. Guo [1] introduced Meyer-König and Zeller

operators as

M̂n(f ,x)=∞∑k=1pn,k+1(x)

(n+k−1)(n+k−3)(n−2)

∫ 10pn−2,k−1(t)f (t)dt, (1.3)

where pn,k(x) is defined in (1.2).Guo [1] obtained the following theorem.

Theorem 1.1. Let f be a function of bounded variation on [0,1], x ∈ (0,1).Then, for all n sufficiently large,

∣∣∣∣M̂n(f ,x)− 12{f(x+)+f (x−)}

∣∣∣∣≤ 7nxn∑k=1Vx+(1−x)/

√k

x−x/√k(gx)

+ 50√n·x3/2

∣∣f (x+)−f (x−)∣∣,(1.4)

http://dx.doi.org/10.1155/S0161171203208218http://dx.doi.org/10.1155/S0161171203208218http://dx.doi.org/10.1155/ijmmshttp://www.hindawi.com

2004 V. GUPTA AND N. KUMAR

where

gx(t)=

f(t)−f (x−), 0≤ t < x,0, t = x,f(t)−f (x+), x < t ≤ 1,

(1.5)

and Vba (gx) is the total variation of gx on [a,b].

Gupta [3] gave a sharp estimate as in the following theorem.

Theorem 1.2. Let f be a function of bounded variation on [0,1], x ∈ (0,1).Then, for all n sufficiently large,

∣∣∣∣M̂n(f ,x)− 12{f(x+)+f (x−)}

∣∣∣∣≤ 7nxn∑k=1Vx+(1−x)/

√k

x−x/√k(gx)

+ 11415√n·x3/2

∣∣f (x+)−f (x−)∣∣.(1.6)

Zeng [9] gave the exact bound for Meyer-König and Zeller basis functions

and claimed to obtain the sharp estimate over the results of Guo [1] and Gupta

[3]. Although the bound obtained in [9] is optimum, the main estimate given by

Zeng [9] for the operators M̂n(f ,x) is not correct. Zeng obtained the followingtheorem.

Theorem 1.3. Let f be a function of bounded variation on [0,1]. Then, forevery x ∈ (0,1) and n sufficiently large,

∣∣∣∣M̂n(f ,x)− 12{f(x+)+f (x−)}

∣∣∣∣≤ 7nxn∑k=1Vx+(1−x)/

√k

x−x/√k(gx)

+ 3√8e·x3/2

∣∣f (x+)−f (x−)∣∣.(1.7)

We may remark here that Theorem 1.3 obtained by Zeng [9] has a major

mistake because the right-hand side does not converge to zero for sufficiently

large n. Also, the remark given before [9, Theorem 4.3] is contradictory to themain theorem [9, Theorem 4.3]. This motivated us to give the correct estimate

for these operators and in this note we give an improved estimate for the rate

of convergence on functions of bounded variation for the operators (1.3).

2. Auxiliary results. In this section, we give certain results, which are nec-

essary to prove the main result.

Lemma 2.1 [7]. Let X1,X2,X3, . . . ,Xn be n independent and identically dis-tributed random variables with zero mean and a finite absolute third moment.

If ρ2 = E(X21) > 0, then

supx∈R

∣∣Fn(x)−Φ(x)∣∣≤ (0.409)�3,n, (2.1)

A NOTE ON INTEGRAL MODIFICATION . . . 2005

where Fn is the distribution function and �3,n is the Liapounov ratio given by

�3,n =(ρ3ρ3/22

), ρ3 = E

(∣∣X1∣∣3). (2.2)

Lemma 2.2 [7]. If {ξi}, i= 1,2,3, . . ., are independent random variables withthe same geometric distribution

P(ξi)= xk(1−x), x ∈ (0,1), i= 1,2,3, . . . , (2.3)

then

E(ξi)= x

1−x , ρ2 = E(ξi−E

(ξi))2 = x

(1−x)2 (2.4)

and ηn =∑ni=1ξi is a random variable with distribution

P(ηn = k

)=(n+k−1

k

)xk(1−x)n. (2.5)

Lemma 2.3 [9]. For all k,n∈N and x ∈ (0,1],

pn,k(x) <1√

2e√nx3/2

, (2.6)

where the constant 1/√

2e is the best possible.

Lemma 2.4. For k≥ 1, x ∈ (0,1), and n> 2,∣∣∣∣∣∣k+1∑j=2pn,j(x)−

k−1∑j=1pn−1,j(x)

∣∣∣∣∣∣≤3√nx3/2

. (2.7)

Proof. By (2.5), we have

pn,k(x)= P(ηn = k

)= P(k−1< ηn ≤ k)

= P(k−1−nx/(1−x)√

nx/(1−x) <ηn−nx/(1−x)√nx/(1−x) ≤

k−nx/(1−x)√nx/(1−x)

).

(2.8)

Using Lemma 2.1, we obtain

∣∣∣∣∣P(ηn = k)− 1√2π∫ (k−nx/(1−x))/(√nx/(1−x))(k−1−nx/(1−x))/(√nx/(1−x))

e−t2/2dt

∣∣∣∣∣< 2(0.409)ρ3√nρ3/22 . (2.9)


Next, we estimate ρ3. For Tr (x) =∑∞k=0krxk(1−x), we have by an easy com-

putation

T0(x)= 1, T1(x)= x1−x , T2(x)=

x(1+x)(1−x)2 ,

T3(x)= x3+4x2+x(1−x)3 , T4(x)=

x4+11x3+11x2+x(1−x)4 .

(2.10)

Also, ifMr(x)=∑∞k=0(k−x/(1−x))rxk(1−x) stands for the central moment

of order r about the mean x/(1−x), it is easily checked by using the abovethat

M2(x)=2∑j=0

(2

j

)(−1)jT2−j(x)

(T1(x)

)j = x(1−x)2 ,

M4(x)=4∑j=0

(4

j

)(−1)jT4−j(x)

(T1(x)

)j = x3+7x2+x(1−x)4 .

(2.11)

Thus, using the inequality ρ3 ≤ (M2(x)M4(x))1/2, we have

ρ3 ≤(

x(1−x)2 ·

x3+7x2+x(1−x)4

)1/2≤ 3(1−x)3 , x ∈ (0,1). (2.12)

So, the right-hand side of (2.9) is less than 2.454/√nx3/2, that is, less than

5/2√nx3/2.

Also, since

k+1∑j=0pn,j(x)= P

(ηn ≤ k+1

),

k−1∑j=0pn−1,j(x)= P

(ηn−1 ≤ k−1

), (2.13)

thus

∣∣∣∣∣∣k+1∑j=0pn,j(x)− 1√

2π

∫ (k+1−nx/(1−x))/(√nx/(1−x))−∞

e−t2/2dt

∣∣∣∣∣∣<5

4√nx3/2

,

∣∣∣∣∣∣k−1∑j=0pn−1,j(x)− 1√

2π

∫ (k−1−(n−1)x/(1−x))/(√(n−1)x/(1−x))−∞

e−t2/2dt

∣∣∣∣∣∣<

5

4√(n−1)x3/2 .

(2.14)


On the other hand,

pn,0(x)+pn,1(x)= o(n−1

). (2.15)

Hence, for n sufficiently large, we have

∣∣∣∣∣∣k+1∑j=2pn,j(x)−

k−1∑j=0pn−1,j(x)

∣∣∣∣∣∣<10

4√nx3/2

+ 1√2π

∫In,ke−t

2/2dt

<5

2√nx3/2

+ 1−x√2πnx

<5

2√nx3/2

+ 12√nx3/2

= 3√nx3/2

,

(2.16)

where

In,k =[k−1−nx/(1−x)√

nx/(1−x) ,k−nx/(1−x)√nx/(1−x)

]. (2.17)

This completes the proof of the lemma.

Lemma 2.5 [1]. For k≥ 1, x ∈ (0,1), and n> 2,

k−1∑j=0pn−1,j(x)= (n+k−2)(n+k−3)(n−2)

∫ 1xpn−2,k−1(t)dt. (2.18)

3. Main result. In this section, we prove the following main theorem.

Theorem 3.1. Let f be a function of bounded variation on [0,1]. Then, forevery x ∈ (0,1) and n sufficiently large,∣∣∣∣M̂n(f ,x)− 12

{f(x+)+f (x−)}

∣∣∣∣≤ 7nxn∑k=1Vx+(1−x)/

√k

x−x/√k(gx)

+(

3+ 1√8e

)1√nx3/2

∣∣f (x+)−f (x−)∣∣,(3.1)

where Vba (gx) is the total variation of gx on [a,b].

Proof. First,

∣∣∣∣M̂n(f ,x)− 12{f(x+)+f (x−)}

∣∣∣∣≤ ∣∣M̂n(gx,x)∣∣+ 1

2

∣∣f (x+)−f (x−)∣∣·∣∣M̂n(sign(t−x),x)∣∣.(3.2)


Now, with the kernel

Kn(x,t)=∞∑k=1

(n+k−2)(n+k−3)(n−2) pn,k+1(x)pn−2,k−1(t), (3.3)

we have

M̂n(sign(t−x),x)=

∫ 10Kn(x,t)sign(t−x)dt

=∫ 1xKn(x,t)dt−

∫ x0Kn(x,t)dt

=An(x)−Bn(x).

(3.4)

Using Lemma 2.5, we have

An(x)=∫ 1xKn(x,t)dt

=∞∑k=1pn,k+1(x)

(n+k−2)(n+k−3)(n−2)

∫ 1xpn−2,k−1(t)dt

=∞∑k=1

pn,k+1(x)

k−1∑j=0pn−1,j(x)

.

(3.5)

Using Lemma 2.4, we get

∣∣∣∣∣∣An(x)−∞∑k=1

pn,k+1(x)

k+1∑j=2pn,j(x)

∣∣∣∣∣∣≤

3√nx3/2

. (3.6)

Next, by Lemma 2.3, we get

∣∣∣∣∣∣∞∑k=1

pn,k+1(x)

k+1∑j=2pn,j(x)

− 1

2

∣∣∣∣∣∣≤1

2√

2e√nx3/2

. (3.7)

Hence,

∣∣M̂n(sign(t−x),x)∣∣= ∣∣An(x)−Bn(x)∣∣= ∣∣2An(x)−1+o(n−1)∣∣≤ 2

(3+ 1√

8e

)1√nx3/2

+o(n−1).(3.8)

Substituting the value of |M̂n(sign(t−x),x)| and proceeding along the linesof [1] for the value of |M̂n(gx,x)|, the theorem follows.

This completes the proof of theorem.

Remark 3.2. We remark here that in order to prove the main theorem, the

inequality ρ3≤16/(1−x)3 is used in Theorem 1.1 and the value ρ3≤3√

3/(1−x)3


is used in Theorem 1.2. Zeng [9] used this value and the exact bound and

gave the misprinted theorem (Theorem 1.3). Although Zeng obtained the exact

bound for Meyer-König and Zeller basis functions, he has not used it correctly

to obtain his main result, that is, Theorem 1.3.

4. Bezier variant of the operators M̂n. In this section, we propose the Beziervariant of the integrated Meyer-König and Zeller operators as

M̂n,α(f ,x)=∞∑k=1Q(α)n,k+1(x)

(n+k−1)(n+k−3)(n−2)

∫ 10pn−2,k−1(t)f (t)dt, (4.1)

where α ≥ 1, Q(α)n,k(x) = (∑∞j=kpn,j(x))α − (

∑∞j=k+1pn,j(x))α, M̂n,α(1,x) = 1.

It is easily verified that the operators (4.1) are linear positive operators. In

particular, for α= 1, operators (4.1) reduce to the operators (1.3).

References

[1] S. S. Guo, Degree of approximation to functions of bounded variation by certainoperators, Approx. Theory Appl. 4 (1988), no. 2, 9–18.

[2] , On the rate of convergence of the integrated Meyer-König and Zeller op-erators for functions of bounded variation, J. Approx. Theory 56 (1989),no. 3, 245–255.

[3] V. Gupta, A sharp estimate on the degree of approximation to functions of boundedvariation by certain operators, Approx. Theory Appl. (N.S.) 11 (1995), no. 3,106–107.

[4] , A note on Meyer-König and Zeller operators for functions of boundedvariation, Approx. Theory Appl. (N.S.) 18 (2002), no. 3, 99–102.

[5] , On a new type of Meyer-Konig and Zeller operators, JIPAM. J. Inequal.Pure Appl. Math. 3 (2002), no. 4, Article 57, 1–10.

[6] V. Gupta and U. Abel, Rate of convergence by a new type of Meyer König andZeller operators, to appear in Fasc. Math.

[7] E. R. Love, G. Prasad, and A. Sahai, An improved estimate of the rate of conver-gence of the integrated Meyer-König and Zeller operators for functions ofbounded variation, J. Math. Anal. Appl. 187 (1994), no. 1, 1–16.

[8] W. Meyer-König and K. Zeller, Bernsteinsche Potenzreihen, Studia Math. 19 (1960),89–94 (German).

[9] X. Zeng, Bounds for Bernstein basis functions and Meyer-König and Zeller basisfunctions, J. Math. Anal. Appl. 219 (1998), no. 2, 364–376.

[10] , Rates of approximation of bounded variation functions by two generalizedMeyer-König and Zeller type operators, Comput. Math. Appl. 39 (2000),no. 9-10, 1–13.

Vijay Gupta: School of Applied Sciences, Netaji Subhas Institute of Technology, AzadHind Fauj Marg, Sector 3 Dwarka, Pappankalan, New Delhi 110 045, India

E-mail address: [email protected]

Niraj Kumar: School of Applied Sciences, Netaji Subhas Institute of Technology, AzadHind Fauj Marg, Sector 3 Dwarka, Pappankalan, New Delhi 110 045, India

E-mail address: [email protected]

mailto:[email protected]:[email protected]

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A NOTE ON INTEGRAL MODIFICATION OF THE MEYER-KÖNIG … · 2019. 8. 1. · OF THE MEYER-KÖNIG AND ZELLER OPERATORS VIJAY GUPTA and NIRAJ KUMAR Received 6 August 2002 Guo (1988) introduced