The generalization of bivariate MKZ operators by multiple generating functions

The generalization of bivariate MKZ operators by multiple generating functions

Fatma Ta¸sdelen

, Ay¸segül Erençin

aAnkara University, Faculty of Science, Department of Mathematics, 06100 Ankara, Turkey bAbant ˙Izzet Baysal University, Faculty of Arts and Sciences, Department of Mathematics, 14280 Bolu, Turkey

Received 30 May 2006 Available online 10 October 2006

Submitted by H.M. Srivastava

Abstract

In the present paper, we study approximation properties of multiple generating functions type bivariate Meyer-König and Zeller (MKZ) operators with the help of Volkov type theorem. We compute the order of convergence of these operators by means of modulus of continuity and the elements of modified Lipschitz class. Finally, we give application to partial differential equations.

©2006 Elsevier Inc. All rights reserved.

Keywords:Positive linear operators; Volkov type theorem; Bivariate Meyer-König and Zeller operators; Modified Lipschitz class; Modulus of continuity

1. Introduction

For a function defined on[0,1)the Meyer-König and Zeller operators [12] and Berstein power series defined by Cheney and Sharma [5] are given by

M_n(f;x)=(1−x)ⁿ⁺¹ ∞ k=0

f k

k+n+1

n+k k

x^k (1.1)

and

* Corresponding author.

E-mail addresses:tasdelen@science.ankara.edu.tr (F. Ta¸sdelen), erencina@hotmail.com (A. Erençin).

0022-247X/$ – see front matter ©2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2006.09.024

M_n^∗(f;x)=(1−x)ⁿ⁺¹ ∞ k=0

f k

k+n

n+k k

x^k, (1.2)

respectively. Recently, some generalizations of these operators were considered in [1,2,6,8,9].

Do˘gru [6] defined a sequence of generalized linear positive operators which includes MKZ op- erators and gave some approximation properties of these operators. Agratini [2] defined another sequence of generalized linear positive operators which includes the sequence obtained in [6]

and proved that the sequence converges to the identity operator. Gupta [8] introduced a integral modification of the Meyer-König and Zeller Bezier type operators and estimated the rate of con- vergence of functions of bounded variation. Gupta [9] also introduced Bezier variant of the MKZ operators and studied the rate of convergence by means of the decomposition technique of func- tions of bounded variation. In [1], Abel, Gupta and Ivan considered a general Durrmeyer variant of the MKZ operators and derived the complete asymptotic expansion for these operators. More recently, Altın, Do˘gru and Ta¸sdelen [4] introduced a generalization of(1.1)and(1.2)by linear generating functions as follows:

L_n(f;x)= 1 hn(x, t )

∞ k=0

f

ak,n

ak,n+bn

C_k,n(t )x^k (1.3)

for 0_a^ak,n

k,n+bn A,A∈(0,1), where{h_n(x, t )}n∈Nis the generating functions for the sequence of functions{C_k,n(t )}k∈N0 in the formh_n(x, t )=_∞

k=0C_k,n(t )x^k for allt∈I that is any subin- terval ofR.Authors studied some approximation properties of the operators(1.3).

In this paper, we consider the sequence of linear positive operators of two variables Ln,m(f;x, y)= 1

Ψ_n,m(x, y, s, t ) ∞ k=0

∞ l=0

f

a_k,n

a_k,n+b_n, c_l,m c_l,m+d_m

Γ_k,l^n,m(s, t )x^ky^l (1.4) for 0 _ak,n^ak,n_+bn Aand 0 _cl,m^cl,m_+dm B; A, B ∈(0,1)where {Ψ_n,m(x, y, s, t )}n,m∈N is the multiple generating functions (see [13]) for the sequence of functions{Γ_k,l^n,m(s, t )}k,l∈N0 of the form

Ψ_n,m(x, y, s, t )=^∞

k=0

∞ l=0

Γ_k,l^n,m(s, t )x^ky^l

withΓ_k,l^n,m(s, t )0 for all(s, t )∈D²⊂R². Assume that the following conditions hold:

(1) Ψ_n,m(x, y, s, t )=(1−x)Ψ_n+1,m(x, y, s, t ), (2) a_k+1,nΓ_k^n,m_+1,l(s, t )=b_nΓ_k,l^n+1,m(s, t ),

(3) a_k+1,n=a_k,n+1+ϕ_n,|ϕ_n|n₁<∞anda_0,n=0, (4) bn→ ∞,^bn_b⁺_n¹ →1 andbn=0 for alln∈N, (5) Ψn,m(x, y, s, t )=(1−y)Ψn,m+1(x, y, s, t ), (6) c_l+1,mΓ_k,l^n,m₊₁(s, t )=d_mΓ_k,l^n,m+1(s, t ),

(7) c_l+1,m=c_l,m+1+φ_m,|φ_m|m₁<∞andc_0,m=0, (8) d_m→ ∞, ^dm_dm⁺¹ →1 andd_m=0 for allm∈N.

Remark 1.Some particular cases of the operators(1.4)are defined as follows:

Case1: If we take

(a) Ψ_n,m(x, y, s, t )=h_n(x, t )h_m(y, s), (b) Γ_k,l^n,m(s, t )=C_k,n(t )C_l,m(s),

then(1.4)becomes a bivariate extension of(1.3).

If also

(c) f (x, y)=f₁(x)f₂(y),

then we haveL_n,m(f;x, y)=L_n(f₁;x)L_m(f₂;y).

Case2: If we chose Ψ_n,m(x, y, s, t )=(1 −x)⁻ⁿ⁻¹(1−y)^−m−1, Γ_k,l^n,m(s, t )=_n+k

k

_l+m

l

, a_k,n=k,b_n=n+1,c_l,m=landd_m=m+1, then as a result of case 1 the operators(1.4) turn to be Meyer-König and Zeller operators of two variables.

Case3: If (c) holds and if we takeΨ_n,m(x, y, s, t ),Γ_k,l^n,m(s, t ),a_k,n,b_nandc_l,mas in case 2 and dm=m, then we getLn,m(f;x, y)=Mn(f1;x)M_m^∗(f2;y).

2. Approximation properties ofLn,m

In this section, we shall investigate the approximation properties ofL_n,mwith the help of the test functions

f₀(u, v)=1, f₁(u, v)= u

1−u, f₂(u, v)= v 1−v, f3(u, v)=

u 1−u

2

+ v

1−v 2

. (2.1)

Because for the nodesu=_ak,n^ak,n_+bn andv=_cl,m^cl,m_+dm the denominators of ₁₋^u_u =^a_b^k,n_n and₁₋^v_v=

cl,m

dm are free ofkandl, respectively.

Now we define the space in this work. Let[0, A] × [0, B] =I²and letH_w(I²)be a space of real valued functionsf ∈C(I²)satisfying

f (u, v)−f (x, y)w

f,

u 1−u, v

1−v

− x

1−x, y 1−y

. (2.2)

In(2.2),|(₁₋^u_u,₁₋^v_v)−(₁₋^x_x,₁₋^y_y)| = (₁₋^u_u−₁₋^x_x)²+(₁₋^v_v−₁₋^y_y)²andwis the modulus of continuity off denoted by forδ >0,

w(f, δ)=supf (u, v)−f (x, y); (u, v), (x, y)∈I²,(u, v)−(x, y)< δ so that the following conditions are satisfied

(i) w(f, δ)is non-negative and increasing forδ, (ii) limδ→0w(f, δ)=0.

It is also well known that for each(u, v)∈I², f (u, v)−f (x, y)w(f, δ)

1+|(u, v)−(x, y)| δ

. (2.3)

Note that the value ofL_n,mat a point(x, y)∈I²byL_n,m(f (u, v);x, y)or simplyL_n,m(f;x, y).

An extension of Korovkin Theorem [10] for linear positive operators of two variables was given by Volkov [14]. We now prove a Volkov type theorem using the test functions(2.1)for the linear positive operators of two variables acting fromH_w(I²)toC(I²)for investigation of the approximation properties of the operators(1.4).

Theorem 1.LetAn,mbe a sequence of linear positive operators acting fromHw(I²)toC(I²) and satisfying the four conditions

n,mlim→∞An,m(fi;x, y)−fi(x, y)

C(I²)=0, i=0,1,2,3, (2.4)

wheref₀, f₁, f₂andf₃are given by(2.1)and.C(I²)denotes the sup norm on the spaceC(I²).

Then for allf ∈H_w(I²), we have

n,mlim→∞An,m(f;x, y)−f (x, y)

C(I²)=0.

Proof. Iff ∈Hw(I²), then using the property (ii) in the definition ofHw(I²), we can write f (u, v)−f (x, y)< ε for

u

1−u− x 1−x

2

+ v

1−v− y 1−y

2

< δ.

Since(u, v), (x, y)∈I²andf ∈C(I²),there exists a positive constantMsuch that f (u, v)−f (x, y)<2M.

For (₁₋^u_u−₁₋^x_x)²+(₁₋^v_v−₁₋^y_y)²δsince(₁₋^u_u−₁₋^x_x)²+(₁₋^v_v−₁₋^y_y)²δ²we can get f (u, v)−f (x, y)<2M

δ² u

1−u− x 1−x

2

+ v

1−v − y 1−y

2 .

Therefore for all(u, v), (x, y)∈I²andf ∈C(I²), we have f (u, v)−f (x, y)< ε+2M

δ² u

1−u− x 1−x

2

+ v

1−v− y 1−y

2

. (2.5)

Using linearity and positivity of the operatorsA_n,mand(2.5)one gets A_n,m(f;x, y)−f (x, y)

A_n,mf (u, v)−f (x, y);x, y

+f (x, y)A_n,m(f0;x, y)−f0(x, y)

ε+

ε+M+2M δ²

An,m(f0;x, y)−f0(x, y)+4M

δ² An,m(f1;x, y)−f1(x, y) +4M

δ² A_n,m(f₂;x, y)−f₂(x, y)+2M

δ² A_n,m(f₃;x, y)−f₃(x, y).

Hence taking supremum over (x, y)∈I² and using the conditions (2.4) the proof is com- pleted. 2

In the light of Theorem 1, we now prove the following main result.

Theorem 2.LetL_n,mbe defined by(1.4). Then for allf ∈H_w(I²)we have

n,mlim→∞L_n,m(f;x, y)−f (x, y)

C(I²)=0.

Proof. It is enough to prove the conditions of Theorem 1 which are

n,mlim→∞L_n,m(f_i;x, y)−f_i(x, y)

C(I²)=0, i=0,1,2,3, (2.6)

wheref₀, f₁, f₂andf₃are given by(2.1).

It is clear that

Ln,m(f0;x, y)=1. (2.7)

By using the conditions (1)–(3), we have L_n,m(f₁;x, y)= 1

b_n

1 Ψ_n,m(x, y, s, t )

∞ k=1

∞ l=0

a_k,nΓ_k,l^n,m(s, t )x^ky^l

= x b_n

1 Ψ_n,m(x, y, s, t )

∞ k=0

∞ l=0

a_k+1,nΓ_k^n,m_+1,l(s, t )x^ky^l

= x

Ψ_n,m(x, y, s, t ) ∞ k=0

∞ l=0

Γ_k,l^n+1,m(s, t )x^ky^l

= x

1−x. (2.8)

In a similar way that of(2.8)by using (5)–(7) and (1)–(8) we obtain L_n,m(f₂;x, y)= y

1−y (2.9)

and

L_n,m(f₃;x, y)=bn+1

bn

x 1−x

2

+ϕn

bn

x

1−x +dm+1

dm

y 1−y

2

+φm

dm

y

1−y, (2.10) respectively. Thus from (2.7)–(2.10) we reach to(2.6)which is the desired result. 2

3. Rates of convergence

In this section, we compute the rates of convergence of L_n,m(f (u, v);x, y) tof (x, y) by means of the modulus of continuity and the elements of modified Lipschitz class.

Theorem 3.If the operatorL_n,mis defined by(1.4), then for allf ∈H_w(I²)we have L_n,m(f;x, y)−f (x, y)

C(I²)

1+K¹²

w(f, δ_nm),

whereK=max{₁₋^A_A, (₁₋^A_A)²,₁₋^B_B, (₁₋^B_B)²}andδ_nm=(^bn_b⁺¹

n +^dm_dm⁺¹ −2+^ϕ_bnⁿ+^φ_dm^m)¹². Proof. By linearity and monotonicity ofL_n,mand using(2.3)we obtain

L_n,m(f;x, y)−f (x, y)

L_n,mf (u, v)−f (x, y);x, y

= 1

Ψn,m(x, y, s, t ) ∞ k=0

∞ l=0

f

a_k,n ak,n+bn

, c_l,m cl,m+dm

−f (x, y)

Γ_k,l^n,m(s, t )x^ky^l

w(f, δnm) Ψn,m(x, y, s, t )

∞ k=0

∞ l=0

1+ (^a_b^k,n

n −₁₋^x_x)²+(^c_d^l,m

m −₁₋^y_y)² δnm

Γ_k,l^n,m(s, t )x^ky^l

=w(f, δ_nm)

1+ 1 δnm

1 Ψn,m(x, y, s, t )

×^∞

k=0

∞ l=0

ak,n

bn − x 1−x

2

+ cl,m

dm − y 1−y

2

Γ_k,l^n,m(s, t )x^ky^l

.

By the Cauchy–Schwarz inequality and Theorem 2, we get L_n,m(f;x, y)−f (x, y)

w(f, δ_nm)

1+ 1 δ_nm

1 Ψ_n,m(x, y, s, t )

∞ k=0

∞ l=0

ak,n

b_n − x 1−x

2

+ c_l,m

d_m − y 1−y

2

Γ_k,l^n,m(s, t )x^ky^{l 1}

2

=w(f, δ_nm)

1+ 1 δ_nm

b_n+1 b_n −1

x 1−x

2

+ d_m+1

d_m −1 y

1−y 2

+ϕ_n bn

x

1−x +φ_m dm

y 1−y

¹

2

.

Thus taking supremum over(x, y)∈I²we obtain the proof of the theorem. 2

We will now study the rates of convergence of the positive linear operatorsLn,mby means of the elements of the modified Lipschitz classLip_M(α)for 0< α1. As a first step we give the definition of the modified Lipschitz class (see [4]) for the functions of two variables. Consider the class of functions defined byLip_M(α),

f (u, v)−f (x, y)M

u 1−u, v

1−v

− x

1−x, y 1−y

^α (3.1)

for (u, v), (x, y)∈I² where M >0 and f ∈C(I²).We can call the class Lip_M(α) as “the modified Lipschitz class.” We point out that Lip_M(α)⊂Lip_M(α).

Theorem 4.LetL_n,mbe given by(1.4). Then for allf ∈Lip_M(α),we have L_n,m(f;x, y)−f (x, y)

C(I²)MK^α2δ_nm^α , whereKandδ_nmare the same as in Theorem3.

Proof. Letf ∈Lip_M(α).By linearity and monotonicity ofLn,mand(3.1)we have L_n,m(f;x, y)−f (x, y)

M

Ψn,m(x, y, s, t ) ∞ k=0

∞ l=0

a_k,n bn − x

1−x 2

+ c_l,m

dm − y 1−y

2^α

2

Γ_k,l^n,m(s, t )x^ky^l.

Applying the Hölder inequality withp=²_α, q=₂₋²_α, we get Ln,m(f;x, y)−f (x, y)

M

1 Ψn,m(x, y, s, t )

∞ k=0

∞ l=0

a_k,n bn − x

1−x 2

+ c_l,m

dm − y 1−y

2

×Γ_k,l^n,m(s, t )x^ky^{l α}

2

.

As in Theorem 3, by using Theorem 2 and taking the supremum over(x, y)∈I²we arrive at the desired result. 2

4. Application to partial differential equations

We consider the operators L^∗_n,m(f;x, y)= 1

Ψn,m(x, y, s, t ) ∞ k=0

∞ l=0

f k

k+bn

, l

l+dm

Γ_k,l^n,m(s, t )x^ky^l (4.1) which is the special case of(1.4)when takinga_k,n=kandc_l,m=l.

In this section by Theorem 5, we shall give a partial differential equation for the bivariate operators (4.1) that seems to be fundamental for the investigation of several kinds of linear positive operators. There are some papers in which differential equations are given for some linear positive operators. For example, see [3,4,6,7,11,15].

Theorem 5.Let

∂

∂x

Ψ_n,m(x, y, s, t )

=K_n(x)Ψ_n,m(x, y, s, t ), (4.2)

∂

∂y

Ψn,m(x, y, s, t )

=Hm(y)Ψn,m(x, y, s, t ) (4.3)

andg(u, v)=₁₋^u_u+₁₋^v_v.Then for each(x, y)∈I²andf ∈H_w(I²),L^∗_n,mas defined in(4.1) satisfies the functional partial differential equation

x bn

∂

∂x + y dm

∂

∂y

L^∗_n,m(f;x, y)= − x

bn

K_n(x)+ y dm

H_n(y)

L^∗_n,m(f;x, y) +L^∗_n,m(f g;x, y).

Proof. Sincef ∈C(I²), the power series on the right side of(4.1)converges onI².Hence we can differentiate partially it term by term inI².So differentiating with respect tox and using (4.2)we have

∂

∂xL^∗_n,m(f;x, y)= −K_n(x)L^∗_n,m(f;x, y)

+ 1

Ψn,m(x, y, s, t ) ∞ k=1

∞ l=0

kf k

k+bn

, l

l+dm

Γ_k,l^n,m(s, t )x^k−1y^l.

Multiplying both sides of this by_b^x

n we obtain

x bn

∂

∂xL^∗_n,m(f;x, y)= − x bn

K_n(x)L^∗_n,m(f;x, y)

+ 1

Ψ_n,m(x, y, s, t ) ∞ k=0

∞ l=0

k b_nf

k k+b_n, l

l+d_m

Γ_k,l^n,m(s, t )x^ky^l.

(4.4) In a similar manner differentiating(4.1)with respect toyand using(4.3),we have

y d_m

∂

∂yL^∗_n,m(f;x, y)= − y

d_mH_m(y)L^∗_n,m(f;x, y)

+ 1

Ψ_n,m(x, y, s, t ) ∞ k=0

∞ l=0

l d_mf

k k+b_n, l

l+d_m

Γ_k,l^n,m(s, t )x^ky^l.

(4.5) Summing(4.4)and(4.5)side by side and using(_b^k

n +_d^l_m)=g(_k+^k_b

n,_l+^l_d

m), it follows that x

b_n

∂

∂x + y d_m

∂

∂y

L^∗_n,m(f;x, y)

= − x

b_nK_n(x)+ y d_mH_m(y)

L^∗_n,m(f;x, y)

+ 1

Ψn,m(x, y, s, t ) ∞ k=0

∞ l=0

f k

k+bn

, l

l+dm

g

k k+bn

, l

l+dm

Γ_k,l^n,m(s, t )x^ky^l.

In view of(4.1),we obtain the required result. 2

Remark 2.ReplaceI²byIⁿ= [0, A1] × · · · × [0, An]whereA₁, . . . , A_n∈(0,1)and consider the modulus of continuityw(f, δ)for the functionsf ofn-variables, given by forδ >0,

w(f, δ)=supf (u₁, . . . , u_n)−f (x₁, . . . , x_n);

(u₁, . . . , u_n), (x₁, . . . , x_n)∈Iⁿ,(u₁, . . . , u_n)−(x₁, . . . , x_n)< δ .

Then letH_w(Iⁿ)be space of all real valued functions satisfying f (u1, . . . , u_n)−f (x1, . . . , x_n)

w

f,

u₁

1−u₁, . . . , u_n 1−u_n

− x₁

1−x₁, . . . , x_n 1−x_n

.

Now consider the class of functions defined byLip_M(α), f (u1, . . . , un)−f (x1, . . . , xn)M

u₁

1−u₁, . . . , u_n 1−u_n

− x₁

1−x₁, . . . , x_n 1−x_n

^α for(u₁, . . . , u_n), (x₁, . . . , x_n)∈Iⁿ, whereM >0, 0< α1 andf ∈C(Iⁿ).

Thus all results in this paper can be extended to the case ofn-variate functions.

References

[1] U. Abel, V. Gupta, M. Ivan, The complete asymptotic expansion for a general Durrmeyer variant of the Meyer-König and Zeller operators, Math. Comput. Modelling (40) (2004) 867–875.

[2] O. Agratini, Korovkin type error estimates for Meyer-König and Zeller operators, Math. Inequal. Appl. 4 (2001) 119–126.

[3] J.A.H. Alkemade, The second moment for the Meyer-König and Zeller operators, J. Approx. Theory 40 (1984) 261–273.

[4] A. Altın, O. Do˘gru, F. Ta¸sdelen, The generalization of Meyer-König and Zeller operators by generating functions, J. Math. Anal. Appl. 312 (2005) 181–194.

[5] E.W. Cheney, A. Sharma, Berstein power series, Canad. J. Math. 16 (1964) 241–253.

[6] O. Do˘gru, Approximation order asymptotic approximation for generalized Meyer-König and Zeller operators, Math.

Balkanica (N.S.) 12 (1998) 359–368.

[7] O. Do˘gru, M.A. Özarslan, F. Ta¸sdelen, On positive operators involving a certain class of generating functions, Studia Sci. Math. Hungar. 41 (4) (2004) 415–429.

[8] V. Gupta, On a new type of Meyer-König and Zeller operators, J. Inequal. Pure Appl. Math. 3 (4) (2002), Art. 57.

[9] V. Gupta, Degree of approximation to function of bounded variation by Bezier variant of MKZ operators, J. Math.

Anal. Appl. 289 (1) (2004) 292–300.

[10] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publish Co., Delhi, 1960.

[11] C.P. May, Saturation and inverse theorem for combinations of a class of exponential-type operators, Canad. J.

Math. 28 (1976) 1224–1250.

[12] W. Meyer-König, K. Zeller, Bersteinsche Potenzreihen, Studia Math. 19 (1960) 89–94.

[13] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Wiley, New York, 1984.

[14] V.I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dokl. Akad. Nauk SSSR (N.S.) 115 (1957) 17–19 (in Russian).

[15] Yu.I. Volkov, On certain positive linear operators, Mat. Zametki 23 (1978) 659–669.