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The Meyer-K önig and Zeller operator is one of the most challenging operators. Sometimes the study of its properties will rely on the weighted approximation by Baskakov operator. In this paper, this relation is extended to complex space; the quantitative estimates and the Voronovskaja type results for analytic functions by complex Meyer-K önig and Zeller operators were obtained.

The well known Meyer-König and Zeller operators are defined for functions f ( x ) ∈ C [ 0,1 ) by [

M n ( f , x ) = ∑ k = 0 ∞ f ( k n + k ) m n , k ( x ) ,

where m n , k ( x ) = ( n + k k ) x k ( 1 − x ) n + 1 .

The Meyer-König and Zeller operators [

The goal of this paper is to extend the results to complex Meyer-König and Zeller operators defined as follows: For analytic functions f : D ¯ R ∪ [ R ,1 ) → C ,0 ≤ R < 1 ,

M n ( f , z ) = ∑ k = 0 ∞ f ( k n + k ) m n , k ( z ) ,

where m n , k ( z ) = ( n + k k ) z k ( 1 − z ) n + 1 , D R = { z ∈ C : | z | < R } .

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

Theorem 1. Suppose that f : D ¯ R ∪ [ R ,1 ) → C is analytic in D ¯ R and continuous in [ R ,1 ) , that is, f ( z ) = ∑ p = 0 ∞ c p z p , for all z ∈ D ¯ R . Let 2 − 1 ≤ r < R < 1 , for all | z | ≤ r and n ≥ 2 , we have

| M n ( f , z ) − f ( z ) | ≤ M r ( f ) n ,

where M r ( f ) = ∑ p = 1 ∞ | c p | ( 2 p ) ! r p − 1 < + ∞ .

Theorem 2. Under the conditions of Theorem 1, for all | z | ≤ r and n ≥ 2 , we have the following Voronovskaja type results

| M n ( f , z ) − f ( z ) − z 2 n f ″ ( z ) | ≤ N r ( f ) n 2 ,

where

1) N r ( f ) = ∑ p = 1 ∞ | c p + 1 | 5 p 2 ( 2 p ) ! r p − 1 < + ∞ , for 2 − 1 ≤ r ≤ 5 − 1 2 ;

2) N r ( f ) = ∑ p = 1 ∞ | c p + 1 | 5 p 2 ( 2 p ) ! r p + 1 < + ∞ , for 5 − 1 2 ≤ r < 1 .

Theorem 3. Under the hypothesis of Theorem 2, if f is not a polynomial of degree ≤ 1 and the series N r ( f ) < + ∞ , then for 2 − 1 ≤ r < R < 1 , we have

‖ M n ( f , z ) − f ( z ) ‖ r ∼ 1 n , n ∈ N ,

here ‖ f ‖ r = sup { | f ( z ) | : z ∈ D r } .

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.

The proof is based on the connection between Meyer-König and Zeller and Baskakov operators. V. Totik was the first to use it [

w 1 ( z ) = w 1 ( α 0 , α 1 , z ) = z α 0 ( 1 − z ) α 1 , z ≠ 0 , 1 , z ∈ D r

defined for real values of the parameters α 0 , α 1 ∈ [ − 1,0 ] . We will utilize the change σ : D r → D l given by

t = σ ( z ) = z 1 − z , z ∈ D r .

Remark 1. σ : D r = { z ∈ C : | z | < r , 0 ≤ r < 1 } → D l = { t ∈ C : | t − t 0 | < l , 0 ≤ l < + ∞ , R e t > − 1 2 } , where l = r 1 − r 2 , t 0 = r 2 1 − r 2 . For example: σ : D 1 2 = { z ∈ C : | z | < 1 2 } → D 2 3 = { t ∈ C : | t − 1 3 | < 2 3 } .

Then, its inverse change σ − 1 : D l → D r is

z = σ − 1 ( t ) = t 1 + t . (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 D l is transformed to a function f defined on D r by τ : g → f

f ( z ) = τ ( g ) ( z ) = λ ( z ) ( g ∘ σ ) ( z ) , λ ( z ) = 1 − z . (2)

The inverse operator τ − 1 transforming a function f defined on D r to a function g defined on D l is τ − 1 : f → g

g ( t ) = τ − 1 ( f ) ( t ) = 1 ( λ ∘ σ − 1 ) ( t ) ( f ∘ σ − 1 ) ( t ) , t ≠ − 1. (3)

When a product of two functions is treated, that means, the associated operator ϒ is defined by

ϒ : w 1 ( z ) = ϒ ( w ) ( z ) = 1 λ ( z ) ( w ∘ σ ) ( z ) , (4)

and its inverse ϒ − 1 is defined by

ϒ − 1 : w ( t ) = ϒ − 1 ( w 1 ) ( t ) = ( λ ∘ σ − 1 ) ( t ) ( w 1 ∘ σ − 1 ) ( t ) .

For f = τ ( g ) , w 1 = ϒ ( w ) , we have

w 1 f = ϒ ( w ) τ ( g ) = ( w ∘ σ ) ( g ∘ σ ) ,

w g = ϒ − 1 ( w 1 ) τ − 1 ( f ) = ( w 1 ∘ σ − 1 ) ( f ∘ σ − 1 ) . (5)

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

Proposition 1. Let F r , F l denote the spaces of all functions defined on D r and D l respectively. Then τ : F l → F r and τ − 1 are linear operators.

Proposition 2. Let w 1 be a weight in D r , w = ϒ − 1 ( w 1 ) ,

F w 1 = { f ∈ F r : w 1 f ∈ L ∞ ( D r ) } ;

F w = { g ∈ F l : w g ∈ L ∞ ( D l ) } .

Then the mapping τ : F w → F w 1 is a linear correspondence with ‖ w 1 τ ( g ) ‖ r = ‖ w g ‖ l , ‖ w τ − 1 ( f ) ‖ l = ‖ w 1 f ‖ r .

Proof. From the definition of the mapping τ (2) and the operator ϒ (4), combining the Proposition 1, we get the mapping τ : F w → F w 1 is a linear correspondence.

Noting that the relation

w 1 τ ( g ) ( z ) = 1 λ ( z ) ( w ∘ σ ) ( z ) ⋅ λ ( z ) ( g ∘ σ ) ( z ) = w ( t ) g ( t ) ,

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 operators

V n ( g , t ) = ∑ k = 0 ∞ g ( k n ) v n , k ( t ) ,

where v n , k ( t ) = ( n + k − 1 k ) t k ( 1 + t ) − n − k .

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

M n ( f , z ) = τ ( V n ( τ − 1 ( f ) ) ) ( z ) , z ∈ D r . (6)

Proof. From the definition of the operator V n ( g , t ) , M n ( f , z ) , Proposition 1 and the identities

n + k n τ ( v n , k ) ( z ) = m n , k ( z ) ,

τ − 1 ( f ) ( k n ) = n + k n f ( k n + k )

valid for k ∈ N ∪ { 0 } , we have (6).

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

‖ w 1 ( M n f − f ) ‖ r = ‖ w ( V n g − g ) ‖ l .

Proof. From Proposition 3, relations ((4), (1), (3)), we obtain for g = τ − 1 f and w = ϒ − 1 w 1 ,

w 1 ( M n f − f ) = ( w ( V n g − g ) ) ∘ σ

and hence

‖ w 1 ( M n f − f ) ‖ r = ‖ w ( V n g − g ) ‖ l .

Remark 3. If the weight w 1 ( z ) = 1 (i.e. α 0 = α 1 = 0 ), the corresponding weight to w 1 ( z ) = 1 is w ( t ) = 1 1 + t .

Then, we have the following auxiliary results.

Lemma 2.1. Under the conditions of Proposition 3, w 1 ( z ) = 1 , w ( t ) = 1 1 + t , we have

‖ M n f − f ‖ r = ‖ w ( t ) ( V n g − g ) ‖ l .

Lemma 2.2. [

T n , p + 1 ( t ) = t ( t + 1 ) n T ′ n , p ( t ) + t T n , p ( t ) .

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 section is the following theorem for upper bound.

Theorem 3.1. Suppose that g : D ¯ L ∪ [ L , + ∞ ) → C is continuous in D ¯ L ∪ [ L , + ∞ ) and analytic in D ¯ L , i.e. g ( t ) = ∑ p = 0 ∞ c p t p . Let 1 2 ≤ l < L < + ∞ , for all | t | ≤ l , n ≥ 2 , we have

‖ w ( t ) ( V n ( g , t ) − g ( t ) ) ‖ l ≤ M l ( g ) n ,

where M l ( g ) = ∑ p = 1 ∞ | c p | ( 2 p ) ! l p − 1 < + ∞ , w ( t ) = 1 1 + t .

Proof. By using the recurrence relation of Lemma 2.2, for all t ∈ C , p = 0 , 1 , 2 , ⋯ , n ≥ 2 , we have

T n , p + 1 ( t ) = t ( t + 1 ) n T ′ n , p ( t ) + t T n , p ( t ) .

From this we immediately get the recurrence formula

w ( t ) ( T n , p ( t ) − t p ) = t n ( T n , p − 1 ( t ) − t p − 1 ) ′ + t [ w ( t ) ( T n , p − 1 ( t ) − t p − 1 ) ] + p − 1 n t p − 1 .

To estimate ‖ w ( t ) ( T n , p ( t ) − e p ( t ) ) ‖ l , we wil use the relation [

| B ′ k ( t ) | ≤ k l ‖ B k ‖ l for all | t | ≤ l , where B k ( t ) is a polynomial of degree ≤ k . Then, we get

‖ w ( t ) ( T n , p ( t ) − e p ( t ) ) ‖ l ≤ l n ‖ T n , p − 1 ( t ) − e p − 1 ( t ) ‖ l p − 1 l + l ‖ w ( t ) ( T n , p − 1 ( t ) − e p − 1 ( t ) ) ‖ l + p − 1 n l p − 1 ,

which implies

‖ w ( t ) ( T n , p ( t ) − e p ( t ) ) ‖ l ≤ ( 3 l ( p − 1 ) n + l ) ‖ w ( t ) ( T n , p − 1 ( t ) − e p − 1 ( t ) ) ‖ l + p − 1 n l p − 1 . (7)

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

‖ w ( t ) ( T n , p ( t ) − e p ( t ) ) ‖ l ≤ ( 2 p ) ! n l p − 1 .

Indeed for p = 1 , ‖ w ( t ) ( T n , 1 ( t ) − e 1 ( t ) ) ‖ l = 0 ≤ 2 n . Suppose that it is true for p > 1 , that is,

‖ w ( t ) ( T n , p ( t ) − e p ( t ) ) ‖ l ≤ ( 2 p ) ! n l p − 1 . (8)

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

‖ w ( t ) ( T n , p + 1 ( t ) − e p + 1 ( t ) ) ‖ l ≤ ( 3 l p n + l ) ( 2 p ) ! n l p − 1 + p n l p .

It remains to prove that for n ≥ 2

( 3 l p n + l ) ( 2 p ) ! n l p − 1 + p n l p ≤ ( 2 ( p + 1 ) ) ! n l p .

By mathematical induction that the last inequality holds true for all p ≥ 1 and n ≥ 2 . From the hypothesis on g, it follows that V n ( g , t ) is analytic in D l , we write

‖ w ( t ) ( V n ( g , t ) − g ( t ) ) ‖ l ≤ ∑ p = 1 ∞ | c p | ⋅ ‖ w ( t ) ( T n , p ( t ) − e p ( t ) ) ‖ l ≤ ∑ p = 1 ∞ | c p | ( 2 p ) ! n l p − 1 .

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

| w ( t ) ( V n ( g , t ) − g ( t ) − t ( 1 + t ) 2 n g ″ ( t ) ) | ≤ N l ( g ) n 2 ,

where

1) for 1 2 ≤ l < L < 1 , N l ( g ) = ∑ p = 1 ∞ | c p + 1 | 5 p 2 ( 2 p ) ! l p − 1 < + ∞ ;

2) for 1 ≤ l < L < + ∞ , N l ( g ) = ∑ p = 1 ∞ | c p + 1 | 5 p 2 ( 2 p ) ! l p + 1 < + ∞ .

Proof. Case I. For 1 2 ≤ l < L < 1 , noting that e p ( t ) = t p , p = 0 , 1 , 2 , ⋯ and T n , p ( t ) = V n ( e p , t ) and V n ( g , t ) = ∑ p = 0 ∞ c p V n ( e p , t ) , we have

| w ( t ) ( V n ( g , t ) − g ( t ) − t ( 1 + t ) 2 n g ″ ( t ) ) | ≤ ∑ p = 1 ∞ | c p | | w ( t ) ( T n , p ( t ) − e p ( t ) − p ( p − 1 ) ( 1 + t ) 2 n t p − 1 ) | .

Using the recurrence relation of Lemma 2.2, we write

T n , p + 1 ( t ) = t ( t + 1 ) n T ′ n , p ( t ) + t T n , p ( t ) .

Denote that

E n , p ( t ) = T n , p ( t ) − e p ( t ) − p ( p − 1 ) ( 1 + t ) 2 n t p − 1 .

Noting that T n , 1 ( t ) − e 1 ( t ) = 0 , for p ≥ 2 , we have

E ′ n , p ( t ) = n t ( 1 + t ) T n , p + 1 ( t ) − n 1 + t T n , p ( t ) − p t p − 1 − p 2 ( p − 1 ) 2 n t p − 1 − p ( p − 1 ) 2 2 n t p − 2 .

By simple computation, we get

E n , p + 1 ( t ) = t ( 1 + t ) n E ′ n , p ( t ) + t E n , p ( t ) + p 2 ( p − 1 ) ( 1 + t ) 2 n 2 t p + p ( p − 1 ) 2 ( 1 + t ) 2 n 2 t p − 1 .

Thus, for all p , n ∈ N , | t | < l , 1 2 ≤ l < L < 1 , we have

| w ( t ) E n , p + 1 ( t ) | ≤ l ( 1 + l ) n | w ( t ) E ′ n , p ( t ) | + l | w ( t ) E n , p ( t ) | + 2 p 3 n 2 l p − 1 . (9)

Using the estimate in the proof of Theorem 3.1, for all p ∈ N , n ≥ 2 and 1 2 ≤ l < L < 1 , we have

‖ w ( t ) ( T n , p ( t ) − e p ( t ) ) ‖ l ≤ ( 2 p ) ! n l p − 1 .

Now we shall estimate

thus,

we obtain step by step following

which follows that

where

Case 2. For

and the relation (10) should be changed to

then,

The Proof of Theorem 1. Combining Lemma 2.1 and Theorem 3.1, we can obtain Theorem 1.

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

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

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

Proof. For all

Applying the inequality

Since f is not a polynomial of degree

Now by Theorem 2, for

Choose

which implies for all

For

i.e.

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

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

We thank the Editor and the referee for their comments. The work is partially supported by NSF of China (11571089, 11871191) and NSF of Hebei Province (2012205028; ZD2019053). The project supported by science foundation of Hebei Normal University.

The authors declare no conflicts of interest regarding the publication of this paper.

Qi, Q.L. and Ma, J.S. (2020) Approximation by Complex Meyer-König and Zeller Operators. Advances in Pure Mathematics, 10, 1-11. https://doi.org/10.4236/apm.2020.101001