Approximation by Complex Meyer König and Zeller Operators

https://www.scirp.org/journal/apm ISSN Online: 2160-0384

ISSN Print: 2160-0368

DOI: 10.4236/apm.2020.101001 Jan. 7, 2020 1 Advances in Pure Mathematics

Approximation by Complex Meyer-König

and Zeller Operators

Qiulan Qi, Jianshuo Ma

School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, China

Abstract

The Meyer-König and Zeller operator is one of the most challenging opera-tors. Sometimes the study of its properties will rely on the weighted approxi-mation by Baskakov operator. In this paper, this relation is extended to com-plex space; the quantitative estimates and the Voronovskaja type results for analytic functions by complex Meyer-König and Zeller operators were ob-tained.

Keywords

Complex Meyer-König and Zeller Operators, Complex Baskakov Operators, Voronovskaja Type Result, Analytic Function

1. Introduction

The well known Meyer-König and Zeller operators are defined for functions

( )

[

0,1

)

f x ∈C by [1]-[7]

(

)

,

( )

0

, ,

n n k

k

k

M f x f m x

n k

∞

=

 

= _{ }

+

 

∑

where

( )

(

)

1

, 1

n k n k

n k

m x x x

k

+

+

 

=_{ } −

  .

The Meyer-König and Zeller operators [1]-[7], the Durrmeyer-type [8]-[15]

have been the object of several investigations in approximation theory. The es-timation of moments, the direct and inverse approximation propertieswere stu-died. Recently, many new modified types [12]-[19] have been constructed for different function spaces. Gal, Mahmudov, Opris etc. [16] [17] [18] [19] ob-tained the quantitative approximation estimates by complex Bernstein-type, Szász-type operators in compact disks.

The goal of this paper is to extend the results to complex Meyer-König and Zeller How to cite this paper: Qi, Q.L. and Ma,

J.S. (2020) Approximation by Complex Meyer-König and Zeller Operators. Ad-vances in Pure Mathematics, 10, 1-11.

https://doi.org/10.4236/apm.2020.101001

Received: November 25, 2019 Accepted: January 4, 2020 Published: January 7, 2020

Copyright © 2020 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

DOI: 10.4236/apm.2020.101001 2 Advances in Pure Mathematics

operators defined as follows: For analytic functions f D: R

[

R,1

)

→C, 0≤ <R 1,

(

)

,

( )

0

, ,

n n k

k

k

M f z f m z

n k

∞

=

 

= _{ }

+

 

∑

where

( )

(

)

1

{

}

, 1 , :

n k

n k R

n k

m z z z D z C z R

k

+

+

 

=_{ } − = ∈ <

  .

We will obtain the following estimates for the complex Meyer-König and Zel-ler operators.

Theorem 1. Suppose that f D: R

[

R,1

)

→C is analytic in DR and

conti-nuous in

[

R,1

)

, that is,

( )

0

p p p

f z c z

∞

=

=

∑

_{, for all} z∈D_R. Let 2 1− ≤ < <r R 1,

for all z ≤r and n≥2, we have

(

,

)

( )

r

( )

,

n

M f

M f z f z

n

− ≤

where

( )

( )

1

1

2 ! p

r p

p

M f c p r

∞

−

=

=

∑

< +∞_.

Theorem 2. Under the conditions of Theorem 1, for all z ≤r and n≥2,

we have the following Voronovskaja type results

(

,

)

( )

₂

( )

( )

2 ,

r n

N f

z

M f z f z f z

n ′′ n

− − ≤

where

1)

( )

2

( )

1

1 1

5 2 ! p

r p

p

N f c p p r

∞

− +

=

=

∑

< +∞_{, for} 2 1 5 1

2

r −

− ≤ ≤ _;

2)

( )

2

( )

1

1 1

5 2 ! p

r p

p

N f c p p r

∞

+ +

=

=

∑

< +∞_{, for} 5 1 1

2 r

−

≤ < _.

Theorem 3. Under the hypothesis of Theorem 2, if f is not a polynomial of degree ≤1 and the series Nr

( )

f < +∞, then for 2 1− ≤ < <r R 1, we have

(

)

( )

1

, , ,

n _r

M f z f z n N

n

−  ∈

here f r =sup

{

f z

( )

:z∈Dr

}

.

The paper is organized as the following: In Section 2, we are going to promote the relationship between the Meyer-König and Zeller and Baskakov operators to complex space. In Section 3, we will study the approximation by the complex Baskakov operators. In Section 4, we will give the proof of Theorems 1 - 3. In Section 5, we will give the conclusion of this paper.

2. The Connection between the Complex Meyer-König and

Zeller and Baskakov Operators

DOI: 10.4236/apm.2020.101001 3 Advances in Pure Mathematics

study a transformation τ mapping functions defined on

0

1

: , 0 ,

2

l

D = ∈_t C t−t <l ≤ < +∞l Ret> − _

  into functions defined on

{

: , 0 1

}

r

D = ∈z C z <r ≤ <r . The operator τ will allow us to relate the results

for the complex Baskakov operators to their counterparts for the complex Mey-er-König and Zeller operators. We will consider variables and functions defined on Dr as z f z,

( )

respectively, and their analogs defined on Dl, as the later will be denoted with t g t,

( )

. We consider the weight functions

( )

(

)

0

(

)

1

1 1 0, 1, 1 , 0,1, r

w z =w

α α

z =zα −z α z≠ z∈D

defined for real values of the parameters α α ∈ −0, 1

[

1, 0

]

. We will utilize the

change σ:Dr →Dl given by

( )

, .

1 r

z

t z z D

z

σ

= = ∈

− Remark 1.

{

}

0

1

: : , 0 1 : , 0 ,

2

r l

D z C z r r D t C t t l l Ret

σ = ∈ < ≤ < → = ∈_ − < ≤ < +∞ > − _

  ,

where 2

1

r l

r

= − ,

2

0 2

1 r t

r

=

− . For example:

1 2

2 3

1 1 2

: : :

2 3 3

D z C z D t C t

σ =_ ∈ < _→ = ∈_ − < _

   .

Then, its inverse change 1

:Dl Dr

σ− _{→ is}

( )

1 _.

1

t

z t

t σ−

= =

+ (1)

Remark 2. From the definition of σ and σ−1_{, we have that the change σ}

and σ−1 _{are linear fractional transformations and conformal mappings.}

A function g defined on Dl is transformed to a function f defined on Dr by

:g f

τ →

( )

( )( )

( )(

)( ) ( )

, 1 .

f z =τ g z =λ z gσ z λ z = −z

₍₂₎

The inverse operator

τ

−1 _{transforming a function f defined on} r

D to a

function g defined on Dl is

1

: f g

τ− _→

( )

( )( )

(

)

( )

(

)

( )

1 1

1

1

, 1.

g t f t f t t

t

τ σ

λ σ

− −

−

= =  ≠ −



(3)

When a product of two functions is treated, that means, the associated opera-tor ϒ _{is defined by}

( )

( )( )

_{( ) (}

)( )

1

1

:w z w z w z ,

z σ

λ

ϒ = ϒ = _

(4)

and its inverse ϒ−1 _{is defined by}

( )

( )( )

(

)

( )

(

)

( )

1 1 1 1

1 1

:w t w t

λ σ

t w

σ

t .

− − − −

ϒ = ϒ =  

DOI: 10.4236/apm.2020.101001 4 Advances in Pure Mathematics

( ) ( ) (

)(

)

1 ,

w f = ϒ w τ g = wσ gσ

( ) ( )

(

)(

)

1 1 1 1

1 1 .

wg= ϒ− w

τ

− f = w 

σ

− f 

σ

− (5)

The operators τ and ϒ have the following properties. From the definition (1)-(3), we yield immediately.

Proposition 1. Let Fr, Fl denote the spaces of all functions defined on Dr

and Dl respectively. Then τ:Fl→Fr and

1

τ− _{are linear operators.}

Proposition 2. Let w1 be a weight in Dr,

( )

1 1

w= ϒ− w ,

( )

{

}

1 : 1 ;

w r r

F = f ∈F w f ∈L_∞ D

( )

{

:

}

.

w l l

F = g∈F wg∈L_∞ D

Then the mapping τ:Fw→Fw1 is a linear correspondence with

( )

1 r l

w

τ

g = wg , w

τ

1

( )

f _l w f1 r.

− ₌

Proof. From the definition of the mapping τ (2) and the operator ϒ (4), combining the Proposition 1, we get the mapping τ:Fw→Fw1 is a linear cor-respondence.

Noting that the relation

( )( )

_{( ) (}

)( ) ( )(

)( )

( ) ( )

1

1

,

w g z w z z g z w t g t

z

τ σ λ σ

λ

= _ ⋅ _ =

one can get the desired result.

The following proposition is very important, it gives the connection between the complex Meyer-König and Zeller operators and the complex Baskakov oper-ators

( )

,

( )

0

, ,

n n k

k

k

V g t g v t

n

∞

=  

= _{ }

 

∑

where ,

( )

(

)

1

1 n k

k n k

n k

v t t t

k

− −

+ −

 

=_{ } +

  .

Proposition 3. For every f such that one of the series in (6) is convergent, for every n∈N, we have

(

)

(

(

1

( )

)

)

( )

, , .

n n r

M f z =

τ

V

τ

− f z z∈D (6)

Proof. From the definition of the operator V_n

( )

g t, ,M_n

(

f z,

)

, Proposition 1

and the identities

( )

n k,

( )

n k,

( )

,

n k

v z m z

n τ

+

=

( )

1 k n k k

f f

n n n k

τ−  ₌ +  

   ₊ 

   

valid for k∈N

{ }

0 , we have (6).

Proposition 4. Under the conditions of Proposition 3, we have

(

)

(

)

1 n _r n _l.

w M f −f = w V g−g

DOI: 10.4236/apm.2020.101001 5 Advances in Pure Mathematics

and 1

1 w= ϒ−w ,

(

)

(

(

)

)

1 n n

w M f− f = w V g−g 

σ

and hence

(

)

(

)

1 n _r n _l.

w M f −f = w V g−g

Remark 3. If the weight w z1

( )

=1 (i.e. α0 =α1=0), the corresponding

weight to w z1

( )

=1 is

( )

1 1

w t t

= + .

Then, we have the following auxiliary results.

Lemma 2.1. Under the conditions of Proposition 3, w z1

( )

=1,

( )

1 1

w t t

= + , we have

( )(

)

.

n r n l

M f −f = w t V g−g

Lemma 2.2. [16] Denoting

( )

p p

e t =t and Tn p,

( )

t =V e tn

( )

p, , Tn p,

( )

t is a

polynomial of degree p, p=0,1, 2,, we have the recurrence formula

( ) (

)

( )

( )

, 1 , ,

1

.

n p n p n p

t t

T t T t tT t

n

+

+ ′

= +

3. Weighted Approximation by the Complex Baskakov

Operators

Theorems 1 - 3 will be proved in Section 4 by transferring the corresponding results for the complex Baskakov operators. In this section, we will prove some properties of the complex Baskakov operators. The first main result of this sec-tion is the following theorem for upper bound.

Theorem 3.1. Suppose that g D: L

[

L,+∞ →

)

C is continuous in

[

,

)

L

D  L +∞ and analytic in D_L, i.e.

( )

0

p p p

g t c t

∞

=

=

∑

_{. Let} 1

2≤ < < +∞l L , for

all t ≤l n, ≥2, we have

( ) ( ) ( )

(

n ,

)

_l l

( )

,

M g

w t V g t g t

n

− ≤

where

( )

( )

1

( )

1

1 2 ! ,

1

p

l p

p

M g c p l w t

t ∞

−

=

= < +∞ =

+

∑

.

Proof. By using the recurrence relation of Lemma 2.2, for all

, 0,1, 2, , 2

t∈C p=  n≥ , we have

( ) (

)

( )

( )

, 1 , ,

1

.

n p n p n p

t t

T t T t tT t

n

+

+ ′

= +

From this we immediately get the recurrence formula

( )

(

( )

)

(

( )

1

)

( )

(

( )

1

)

1

, , 1 , 1

1 .

p p p p

n p n p n p

t p

w t T t t T t t t w t T t t t

n n

− − −

− −

−

′ _{ }

− = − + _ − _+

To estimate

( )

(

n p,

( )

p

( )

)

l

DOI: 10.4236/apm.2020.101001 6 Advances in Pure Mathematics

( )

k k l

k

B t B

l

′ ≤ _{for all} t ≤l, where B tk

( )

is a polynomial of degree ≤k. Then, we get

( )

(

( )

( )

)

( )

( )

( )

(

( )

( )

)

,

1

, 1 1 , 1 1

1 1

,

n p p

l

p

n p p _l n p p

l

w t T t e t

l p p

T t e t l w t T t e t l

n l n

− − − − − − − − ≤ − + − + which implies

( )

(

( )

( )

)

(

)

_{( )}

₍

_{( )}

_{( )}

₎

, 1

, 1 1

3 1 1

.

n p p

l

p

n p p

l

w t T t e t

l p p

l w t T t e t l

n n − − − − −   − ≤_ + _ − +  

(7)

We will prove the following relation by mathematical induction with respect to p:

( )

(

( )

( )

)

( )

1 ,

2 !

.

p

n p p

l

p

w t T t e t l

n

−

− ≤

Indeed for p=1,

( )

(

,1

( )

1

( )

)

2 0

n

l

w t T t e t

n

− = ≤ . Suppose that it is true for

1

p> , that is,

( )

(

( )

( )

)

( )

1 ,

2 !

.

p

n p p

l

p

w t T t e t l

n

−

− ≤ (8)

Now for p+1, by the relations ((7), (8)), we have

( )

(

( )

( )

)

( )

1

, 1 1

2 ! 3

.

p p

n p p

l

p

lp p

w t T t e t l l l

n n n

−

+ +

 

− ≤_ + _ +

 

It remains to prove that for n≥2

( )

2 ! 1

(

2

(

1 !

)

)

3

.

p p p p

p

lp p

l l l l

n n n n

− +

 ₊  _{+ ≤}

 

 

By mathematical induction that the last inequality holds true for all p≥1

and n≥2. From the hypothesis on g, it follows that Vn

( )

g t, is analytic in Dl, we write

( ) ( )

(

( )

)

( )

(

( )

( )

)

( )

1 ,

1 1

2 !

, p .

n _l p n p p p

l

p p

p

w t V g t g t c w t T t e t c l

n

∞ ∞

−

= =

− ≤

∑

⋅ − ≤

∑

Theorem 3.2. Under the conditions of Theorem 3.1, let 1

2≤ < < +∞l L , for

all t ≤l n, ≥2, we have the following Voronovskaja type formula

( )

( ) ( ) (

) ( )

( )

2 1

, ,

2

l n

t t N g

w t V g t g t g t

n n +  _′′  − − ≤     where

1) for 1 1

2≤ < <l L ,

( )

( )

2 1

1 1

5 2 ! p

l p

p

N g c p p l

∞

− +

=

DOI: 10.4236/apm.2020.101001 7 Advances in Pure Mathematics

2) for 1≤ < < +∞l L ,

( )

₁ 2

( )

1

1

5 2 ! p

l p

p

N g c p p l

∞

+ +

=

=

∑

< +∞_.

Proof. Case I. For 1 1

2≤ < <l L , noting that

( )

, 0,1, 2,

p p

e t =t p=  and

( )

( )

, ,

n p n p

T t =V e t and

( )

( )

0

, ,

n p p n p

V g t =

∑

∞₌ c V e t , we have

( )

( ) ( ) (

) ( )

( )

( )

( )

(

)(

)

1 , 1 1 , 2 1 1 . 2 n p

p n p p

p

t t

w t V g t g t g t

n

p p t

c w t T t e t t

n ∞ − = +   ′′ − −     − +   ≤  − −   

∑

Using the recurrence relation of Lemma 2.2, we write

( ) (

)

( )

( )

, 1 , ,

1

.

n p n p n p

t t

T t T t tT t

n + + ′ = + Denote that

( )

( )

( )

(

)(

)

1

, ,

1 1

. 2

p

n p n p p

p p t

E t T t e t t

n

−

− +

= − −

Noting that Tn,1

( )

t −e t1

( )

=0, for p≥2, we have

( ) ( ) ( )

( )

1 2

(

)

1

(

)

2 2

, , 1 ,

1 1

.

1 1 2 2

p p p

n p n p n p

p p p p

n n

E t T t T t pt t t

t t t n n

− − −

+

− −

′ = − − − −

+ +

By simple computation, we get

( ) (

)

( )

( )

2

(

)(

)

(

) (

2

)

1

, 1 , , 2 2

1 1 1 1 1

.

2 2

p p

n p n p n p

t t p p t p p t

E t E t tE t t t

n n n

− +

+ − + − +

′

= + + +

Thus, for all p n, ∈N, t <l, 1 1

2≤ < <l L , we have

( )

( )

(

) ( ) ( )

( )

( )

3 1

, 1 , , 2

1 2

.

p

n p n p n p

l l p

w t E t w t E t l w t E t l

n n

− +

+

′

≤ + +

₍₉₎

Using the estimate in the proof of Theorem 3.1, for all p∈N n, ≥2 and 1

1

2≤ < <l L , we have

( )

(

( )

( )

)

( )

1 ,

2 !

.

p

n p p

l

p

w t T t e t l

n

−

− ≤

Now we shall estimate w t E

( )

n p′,

( )

t for p≥2. Noting that En p,

( )

t is a

polynomial of degree ≤ p, combining the Bernstein's inequality, we have

( )

( )

( )

( )

( )

(

( )

( )

)

( ) (

)(

)

( )

( )

, , 1 , 2

1 1 2

1 1

2 ! 2 2 !

,

n p n p _l

p

n p p

l

l

p p p

p

w t E t w t E t

l

p p t

p

w t T t e t w t t

l n

p p p

p p

l l l

l n n n

DOI: 10.4236/apm.2020.101001 8 Advances in Pure Mathematics

thus,

(

_{) ( ) ( )}

( )

1

, 2

1 4 2 !

,

p n p

l l p p

w t E t l

n n

− +

′ ≤

( )

( )

( )

1

( )

( )

3 1

, 1 2 , 2

4 2 ! 2

,

p p

n p n p

p p p

w t E t l l w t E t l

n n

− −

+ ≤ + +

(10)

( )

( )

( )

1

( )

( )

, 1 2 ,

5 2 !

,

p

n p n p

p p

w t E t l l w t E t

n

−

+ ≤ +

we obtain step by step following

( )

( )

1

( )

2

( )

1

, 1 2 2

1

5 2 ! 5

2 ! ,

p p

p n p

j

p p

pl

w t E t j l

n n

−

− +

=

≤

∑

≤

which follows that

( )

( ) ( ) (

) ( )

( )

2 1

, ,

2

l n

t t N g

w t V g t g t g t

n n

+

 _′′ 

− − ≤

 

 

where

( )

(

)

2

(

)

2

2

5 1 2 1 ! p

l p

p

N g c p p l

∞

−

=

=

∑

− _ − _ < +∞_.

Case 2. For 1≤ <l L, in the proof of Case 1, the relation (9) should be changed to

( )

( )

(

) ( ) ( )

( )

( )

3 1

, 1 , , 2

1 2

,

p

n p n p n p

l l p

w t E t w t E t l w t E t l

n n

+ +

+ _′

≤ + +

and the relation (10) should be changed to

( )

( )

( )

( )

( )

( )

_{( )}

_{( )}

3

1 1

, 1 2 , 2

1

, 2

4 2 ! 2

5 2 !

,

p p

n p n p

p

n p

p p p

w t E t l l w t E t l

n n

p p

l l w t E t

n

− +

+

+

≤ + +

≤ +

then,

( )

2

( )

1

1 1

5 2 ! p .

l p

p

N g c p p l

∞

+ +

=

=

∑

< +∞

4. The Proof of Theorems 1 - 3

The Proof of Theorem 1. Combining Lemma 2.1 and Theorem 3.1, we can ob-tain Theorem 1.

The Proof of Theorem 2. From Lemma 2.1 and Theorem 3.2, we have Theo-rem 2.

In what follows we obtain the exact degree in the approximation by

(

,

)

n

M f z .

Theorem 4.1. Suppose that the hypothesis on the function f and Theorem 2. If f is not a polynomial of degree ≤1 and the series Nr

( )

f < +∞, then

( )

r

n r

C f

M f f

n

− ≥ _{holds, where} C_r

( )

f depends only on f and r.

DOI: 10.4236/apm.2020.101001 9 Advances in Pure Mathematics

(

)

( )

1

( )

1 2

(

)

( )

( )

, , .

n n

z

M f z f z zf z n M f z f z f z

n n n

 _′′  _′′ 

− = _ + _ − − _

 

 

Applying the inequality F+G r≥ F r− Gr ≥ F r− G r, we obtain

(

)

( )

1

( )

1 2

(

)

( )

( )

, , .

n _{r r} n

r z

M f z f z zf z n M f z f z f z

n n n

 _{ } 

′′ ′′

− ≥ _ − _ − − _{ }

 

 

Since f is not a polynomial of degree ≤1 in DR, we get zf′′

( )

z _r>0. In-deed, supposing the contrary, it follows that zf′′

( )

z =0 for all z ≤r, which

implies f′′

( )

z =0 for all z∈DR\ 0

{ }

. Since f is analytic in DR, this means that f′′

( )

z =0 for all z∈D_R, that is f is a polynomial of degree ≤1, a contra-diction with the hypothesis.

Now by Theorem 2, for Nr

( )

f < +∞, we have

(

,

)

( )

( )

( )

0

(

)

. 2

r n

N f

z

n M f z f z f z n

n ′′ n

− − ≤ → → ∞

Choose n1, such that for all n≥n1, we have

( )

(

)

( )

( )

1

( )

, ,

2

n

r r

r z

zf z n M f z f z f z zf z

n

 

′′ − _ − − ′′_ ≥ ′′

 

which implies for all n≥n1,

(

)

( )

1

( )

, .

2

n _{r r}

M f z f z zf z

n ′′

− ≥

For 1≤ ≤ −n n1 1, we have

(

)

( )

1

(

)

( )

,

( )

, , n r 0,

n _r n _r

C f

M f z f z n M f z f z

n n

− ≥ − = >

i.e. n

(

,

)

( )

_r r

( )

C f

M f z f z

n

− ≥ _{, here}

( )

1,

( )

2,

( )

₁ 1,

( )

( )

1

min , , , ,

2

r r r n r _r

C f = C f C f C − f zf′′ z 

  .

The Proof of Theorem 3. From Lemma 2.1, Theorem 4.1 and Theorem 1, we can obtain Theorem 3.

5. Conclusion

In this paper, the properties of approximation are studied by using the general relation between the Meyer-König and Zeller and Baskakov operators. The geo-metric properties (the shap-preserving) of such complex operators still remain to be studied.

Acknowledgements

He-DOI: 10.4236/apm.2020.101001 10 Advances in Pure Mathematics

bei Normal University.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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