Home On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
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On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks

  • Danshegn Yu EMAIL logo and Zhaojun Pang
Published/Copyright: January 21, 2025
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this study, we introduce a kind of Stancu variant of the complex Bernstein-Durrmeyer-type operators in movable compact disks. Their approximation properties for analytic functions in the movable compact disks are investigated.

Keywords: complex Bernstein-Durrmeyer-type operators; movable compact disks; approximation rates
MSC 2010: 30E10; 41A25

1 Introduction

For any f ∈ C [ 0 , 1 ] , Stancu [1] introduced the following so-called Bernstein-Stancu operators:

(1) B n , α , β ( f ; x ) = ∑ k = 0 n f k + α n + β p n , k ( x ) , x ∈ [ 0 , 1 ] ,

where α and β are two given real numbers satisfying 0 ≤ α ≤ β , and p n k ( x ) are the Bernstein basis functions, i.e.,

p n , k ( x ) = n k x k ( 1 − x ) n − k , k = 0 , 1 , … , n .

When α = β = 0 , B n , α , β ( f ) reduces to the classical Bernstein operators. Both Bernstein operators and Bernstein-Stancu operators have many excellent approximation properties. For example, both types of operators can uniformly approximate continuous functions on [ 0 , 1 ] and have special properties in maintaining the monotonicity, convexity, and some other geometric properties of the objective function.

There are many ways to generalize the Bernstein-Stancu operators. Among them, Gadjiev and Ghorbanalizadeh [2] introduced the following Bernstein-Stancu operators by using shifted knots as follows:

(2) S n , α , β ( f ; x ) = n + β 2 n n ∑ k = 0 n f k + α 1 n + β 1 q n , k ( x ) ,

where x ∈ A n ≔ α 2 n + β 2 , n + α 2 n + β 2 and

q n , k ( x ) = n k x − α 2 n + β 2 k n + α 2 n + β 2 − x n − k , k = 0 , 1 , … , n ,

with α k , β k , k = 1 , 2 are real numbers satisfying 0 ≤ α 1 ≤ β 1 , 0 ≤ α 2 ≤ β 2 . When α 2 = β 2 = 0 , S n , α , β ( f ; x ) reduces to Bernstein-Stancu operators defined in (1); when α 1 = α 2 = β 1 = β 2 = 0 , it reduces to the classical Bernstein operators. Some generalizations of the operators S n , α , β ( f ) and their approximation properties can be found in [3–18].

Approximation properties of complex Bernstein polynomials in compact disks were initially studied by Bernstein [19]. Let f : G → C be an analytic function in the open set G ⊂ C , with D ¯ 1 ⊂ G (where D 1 ≔ { z ∈ C : ∣ z ∣ < 1 } ). Bernstein proved (see [19], for example) that the complex Bernstein polynomials B n ( f , z ) ≔ ∑ k = 0 n n k z k ( 1 − z ) n − k f k n , z ∈ C converge to f uniformly in D ¯ 1 . Exact quantitative estimates and quantitative Voronovaskaja-type results for Bernstein polynomials can be found in [20,21]. Many authors have further investigated the properties of approximation for analytic functions on compact disks by Bernstein polynomials and their generalizations such as Bernstein-Stancu polynomials, Kantorovich-type Bernstein-Stancu polynomials, Durrmeyer-type Bernstein-Stancu polynomials [20–35].

Jiang and Yu [29] introduced a new type of complex Bernstein-Stancu polynomials in movable compact disks. Assume that α 1 , α 2 , β 1 , and β 2 are arbitrary given numbers satisfying 0 ≤ α 1 ≤ β 1 , α 2 ∈ C , and ∣ α 2 ∣ ≤ β 2 . Jiang and Yu [29] defined the new complex Bernstein-Stancu polynomials S n , α , β ( f , z ) :

S n , α , β ( f , z ) ≔ n + β 2 n n ∑ k = 0 n q n , k ( z ) f k + α 1 n + β 1 ,

and the corresponding Kantorovich variant of S n , α , β ( f , z ) :

K n , α , β ( f , z ) ≔ ( n + β 1 + 1 ) n + 1 + β 2 n + 1 n ∑ k = 0 n q n k * ( z ) ∫ k + α 1 n + β 1 + 1 k + α 1 + 1 n + β 1 + 1 f ( t ) d t ,

where

q n , k ( z ) ≔ n k z − α 2 n + β 2 k n + α 2 n + β 2 − z n − k , k = 0 , 1 , … , n , q n k * ( z ) ≔ n k z − α 2 n + β 2 + 1 k n + α 2 + 1 n + β 2 + 1 − z n − k , k = 0 , 1 , … , n .

Obviously, S n , α , β ( f , z ) is a complex variant of the operators defined in (2). In the case when α 2 = β 2 = 0 , S n , α , β ( f , z ) and K n , α , β ( f , z ) are the classical complex Bernstein-Stancu polynomials and the classical Kantrovich-Stancu polynomials, which have been studied well (see, e.g., [21–24]). Jiang and Yu [29] obtained the quantitative estimation of approximation by S n , α , β ( f , z ) and K n , α , β ( f , z ) for analytic functions in the movable compact disks.

Recently, Jiang and Yu [31] introduced the Durrmeyer variant of the complex Bernstein-Stancu polynomials as follows:

S ˜ n , α , β ( f , z ) = n + β 2 n 2 n + 1 ∑ k = 0 n q n k ( z ) ( n + 1 ) ∫ A n q n k ( t ) f n t + α 1 n + β 1 d t .

When α 1 = α 2 = β 1 = β 2 = 0 , S ˜ n , α , β ( f , z ) reduces to the classical Bernstein-Durrmeyer polynomials. The quantitative estimation of approximation by S ˜ n , α , β ( f ) for analytic functions in the movable compact disks was obtained in [31].

As we know, there are many types of generalizations of the classical Bernstein-Durrmeyer operators in real variable case. It is interesting to find the complex variant of these operators and investigate the approximation properties of the variant. Now, we recall a kind of Bernstein-Durrmeyer operator defined by Gupta in [36] as follows:

(3) U n ( f , x ) ≔ n ∑ k = 0 n − 1 p n , k ( x ) ∫ 0 1 p n − 1 , k ( t ) f ( t ) d t + f ( 1 ) p n , n ( x ) .

Gupta studied the rate of approximation by U n ( f ) for functions of bounded variation. These operators are capable of providing better approximation for functions of bounded variation than the usual Durrmeyer operators. The main goal of this study is to construct a new kind of complex Stancu variant of U n ( f ) (see U ¯ n , α , β ( f ) in (4) in Section 2) and to investigate their approximation properties for analytic functions in movable compact disks.

2 Main results

In what follows, we always denote by D R the disk with center O ( 0 , 0 ) and radius R for any fixed R > 1 , i.e., D R ≔ { z ∈ C : ∣ z ∣ < R } . We always assume that α 1 , α 2 , β 1 , and β 2 are arbitrary given numbers satisfying 0 ≤ α 1 ≤ β 1 , 0 ≤ α 2 ≤ β 2 .

Suppose that f : D R → C is analytic in D R , i.e., f ( z ) = ∑ m = 0 ∞ c m z m for all z ∈ D R . Define the new Stancu variant of Bernstein-Durrmeyer-type operators defined in (3) as follows:

(4) U ¯ n , α , β ( f ; z ) = n + β 2 n 2 n n ∑ k = 0 n − 1 q n , k ( z ) ∫ A n q n − 1 , k ( t ) f n t + α 1 n + β 1 + n + β 2 n n z − α 2 n + β 2 n f n 2 + n α 1 + n α 2 + α 1 β 2 ( n + β 1 ) ( n + β 2 ) ,

where A n ≔ α 2 n + β 2 , n + α 2 n + β 2 .

Now, we can give our main result of this study:

Theorem 1

Let R and r be the constants such that 1 ≤ r ≤ R . Suppose that f : D R → C is analytic in D R , i.e., f ( z ) = ∑ m = 0 ∞ c m z m for all z ∈ D R . Then, for any sufficient large n ∈ N such that r 1 + β 2 n ≕ r ¯ < R , we have

∣ U ¯ n , α , β ( f ; z ) − f ( z ) ∣ ≤ K r , n ( α , β ) ( f ) ,

for all z − α 2 n + β 2 ≤ r , where

K r , n ( α , β ) ( f ) ≔ ∑ m = 0 ∞ ∣ c m ∣ m β 2 n r ¯ m + 2 m β 1 n + β 1 r m + m α 2 n + β 2 r ¯ m − 1 + 2 m 2 n r ¯ m + 2 m β 2 n + β 2 r ¯ m .

Also, if 1 ≤ r < r 1 < R , then for any sufficiently large n ∈ N such that r ¯ < R , we have

∣ U ¯ n , α , β ( p ) ( f ; z ) − f ( p ) ( z ) ∣ ≤ K r 1 , n ( α , β ) ( f ) p ! r 1 ( r 1 − r ) p + 1 ,

for all z − α 2 n + β 2 ≤ r 1 , n , p ∈ N .

Remark 1

We investigated the approximation properties of U ¯ n , α , β ( f ) for analytic functions in the compact disk z ∈ C : z − α 2 n + β 2 ≤ r . It should be noted that the compact disk z ∈ C : z − α 2 n + β 2 ≤ r moves with the changing of the parameters n , α 2 and β 2 . Taking α 2 = β 2 = 0 , we have the following approximation results of U ¯ n , α , β ( f ) in the usual compact disks { z ∈ C : ∣ z ∣ ≤ r } .

Corollary 1

Let R and r be the constants such that 1 ≤ r ≤ R . Suppose that f : D R → C is analytic in D R , i.e., f ( z ) = ∑ m = 0 ∞ c m z m for all z ∈ D R . Then, we have

∣ U ¯ n , α , β ( f ; z ) − f ( z ) ∣ ≤ H r , n ( α , β ) ( f ) ,

for all ∣ z ∣ ≤ r , where

H r , n ( α , β ) ( f ) ≔ ∑ m = 0 ∞ ∣ c m ∣ 2 m β 1 n + β 1 + 2 m 2 n r m .

Also, if 1 ≤ r < r 1 < R , then

∣ U ¯ n , α , β ( p ) ( f ; z ) − f ( p ) ( z ) ∣ ≤ H r 1 , n ( α , β ) ( f ) p ! r 1 ( r 1 − r ) p + 1 ,

for all ∣ z ∣ ≤ r 1 , n , p ∈ N .

Remark 2

Gal and Iancu [37] investigated Gruss and Gruss-Voronovskaya-type estimates for complex convolution polynomial operators. It is of interesting to consider whether the method of [37] can be applied to our new operators in movable compact disks.

3 Proof of the result

3.1 Auxiliary lemma

Lemma 1

Let p , n ∈ N ∪ 0 , r ≥ 1 , r ¯ < R . Then, for any z − α 2 n + β 2 ≤ r , we have

∣ U ¯ n , α , β ( e p ; z ) ∣ ≤ r ¯ p ,

where e p ( z ) ≔ z p , p = 0 , 1 , 2 , … .

Proof

Set

U n , α , β * ( f ; z ) = n + β 2 n 2 n n ∑ k = 0 n − 1 q n , k ( z ) ∫ A n q n − 1 , k ( t ) f ( t ) + n + β 2 n n z − α 2 n + β 2 n f n + α 2 n + β 2 .

By simple calculations, we have

U ¯ n , α , β ( e m ; z ) = ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m U n , α , β * ( e p ; z ) .

By noting that

∫ A n q n − 1 , k ( t ) t p d t = ∫ A n n − 1 k t − α 2 n + β 2 k n + α 2 n + β 2 − t n − 1 − k t − α 2 n + β 2 + α 2 n + β 2 p d t = ∫ A n n − 1 k ∑ j = 0 p p j α 2 n + β 2 p − j t − α 2 n + β 2 k + j n + α 2 n + β 2 − t n − 1 − k d t ,

and setting u = n + β 2 n t − α 2 n , we have

∫ A n q n − 1 , k ( t ) t p d t = ∫ 0 1 n − 1 k ∑ j = 0 p p j α 2 n + β 2 p − j n n + β 2 n + j u k + j ( 1 − u ) n − 1 − k d u .

Thus,

U n , α , β * ( e p ; z ) = ∑ j = 0 p p j α 2 n + β 2 p − j n n + β 2 j n ! ( n + j ) ! n + β 2 n n ∑ k = 0 n − 1 q n , k ( z ) F j ( k ) + n + β 2 n n z − α 2 n + β 2 n n + α 2 n + β 2 p = ∑ j = 0 p p j α 2 n + β 2 p − j n n + β 2 j n ! ( n + j ) ! n + β 2 n n ∑ k = 0 n q n , k ( z ) F j ( k ) ,

where F 0 ( k ) = 1 , F j ( k ) = ∏ s = 1 j ( k + s ) , k ≥ 0 , j ≥ 1 , which means that

U n , α , β * ( e p ; z ) = ∑ j = 0 p p j α 2 n + β 2 p − j n n + β 2 j n ! ( n + j ) ! ∑ k = 0 min { n , p } n k Δ 1 k F j ( 0 ) z − α 2 n + β 2 k n + β 2 n k ,

where Δ 1 k F j ( k ) is the finite difference of order k of F j ( k ) with step 1 at the point k .

Now, using the identity,

n + α 2 n + β 2 p = U n , α , β * e p ; n + α 2 n + β 2 = ∑ j = 0 p p j α 2 n + β 2 p − j n n + β 2 j n ! ( n + j ) ! ∑ k = 0 min { n , p } n k Δ 1 k F j ( 0 ) ,

we observe that

∑ j = 0 p p j α 2 n + β 2 p − j n n + β 2 j n ! ( n + j ) ! ∑ k = 0 min { n , p } n k Δ 1 k F j ( 0 ) ≤ 1 .

Note that Δ 1 k F j ( 0 ) ≥ 0 ; thus, for all z − α 2 n + β 2 ≤ r , we have

∣ U n , α , β * ( e p ; z ) ∣ ≤ ∑ j = 0 p p j α 2 n + β 2 p − j n n + β 2 j n ! ( n + j ) ! ∑ k = 0 min { n , p } n k Δ 1 k F j ( 0 ) r k n + β 2 n k ≤ r p n + β 2 n p = r ¯ p .

Thus,

∣ U ¯ n , α , β ( e p ; z ) ∣ ≤ ∑ j = 0 p p j n j α 1 p − j ( n + β 1 ) p r ¯ p = n + α 1 n + β 1 p r ¯ p ≤ r ¯ p .

Lemma 1 is proved.□

3.2 Proof of Theorem 1

It is easy to observe that

∣ U ¯ n , α , β ( f ; z ) − f ( z ) ∣ ≤ ∑ m = 0 ∞ ∣ c m ∣ ∣ U ¯ n , α , β ( e m ; z ) − e m ( z ) ∣ .

Case 1. 0 ≤ m ≤ n . By simple calculation, we obtain U ¯ n , α , β ( e 0 ; z ) = e 0 ( z ) = 1 . Therefore, we only need to consider the case when m ≥ 1 . Write

C p , j = p j α 2 n + β 2 p − j n n + β 2 j n ! ( n + j ) ! .

Then,

∣ U ¯ n , α , β ( e m , z ) − e m ( z ) ∣ ≤ ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m ∑ j = 0 p C p , j ∑ k = 0 p n k Δ 1 k F j ( 0 ) z − α 2 n + β 2 k × n + β 2 n k − 1 + ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m ∑ j = 0 p C p , j ∑ k = 0 p n k Δ 1 k F j ( 0 ) × z − α 2 n + β 2 k − z − α 2 n + β 2 p + ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m × z − α 2 n + β 2 p − z − α 2 n + β 2 m + z − α 2 n + β 2 m − z m ≕ I 1 + I 2 + I 3 + I 4 .

For I 1 , by the inequality

∑ j = 0 p C p , j ∑ k = 0 p n k Δ 1 k F j ( 0 ) ≤ 1 ,

we have

I 1 ≤ ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m r p n + β 2 n p − 1 ≤ ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m r p p β 2 n 1 + β 2 n p − 1 ≤ m β 2 n r ¯ m .

For I 2 , we have

I 2 ≤ ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m ∑ j = 0 p C p , j ∑ k = 0 p − 1 n k Δ 1 k F j ( 0 ) z − α 2 n + β 2 k + ∑ j = 0 p C p , j n p Δ 1 p F j ( 0 ) z − α 2 n + β 2 p − z − α 2 n + β 2 p ≤ ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m ∑ j = 0 p C p , j ∑ k = 0 p − 1 n k Δ 1 k F j ( 0 ) r k + ∑ j = 0 p C p , j n p Δ 1 p F j ( 0 ) − 1 r p ≤ ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m r p ∑ j = 0 p C p , j ∑ k = 0 p n k Δ 1 k F j ( 0 ) − ∑ j = 0 p C p , j n p Δ 1 p F j ( 0 ) + ∑ j = 0 p C p , j n p Δ 1 p F j ( 0 ) − 1 ≤ ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m 2 r p 1 − ∑ j = 0 p C p , j n p Δ 1 p F j ( 0 ) .

Since

∑ j = 0 p C p , j n p Δ 1 p F j ( 0 ) = n n + β 2 p n ! ( n + p ) ! n p p ! = n n + β 2 p ∏ j = 1 p n + j − p n + j ,

then

1 − ∑ j = 0 p C p , j n p Δ 1 p F j ( 0 ) ≤ ∑ j = 1 p 1 − n n + β 2 n + j − p n + j ≤ ∑ j = 1 p β 2 n + β 2 + p n + j ≤ p β 2 n + β 2 + p 2 n ,

where in the first inequality, we used the following inequality:

1 − ∏ j = 1 k x j ≤ ∑ j = 1 k ( 1 − x j ) , 0 ≤ x j ≤ 1 , k = 1 , 2 , … .

Therefore,

I 2 ≤ 2 m 2 n r ¯ m + 2 m β 2 n + β 2 r ¯ m .

For I 3 , we have

I 3 ≤ ∑ p = 0 m m p n p α 1 m − p ( n + β 1 ) m z − α 2 n + β 2 p + n n + β 1 m − 1 z − α 2 n + β 2 m ≤ r m n + α 1 n + β 1 m − n n + β 1 m + n n + β 1 m − 1 r m ≤ 2 r m 1 − n n + β 1 m ≤ 2 m β 1 n + β 1 r m .

For I 4 , we have

I 4 = z − α 2 n + β 2 + α 2 n + β 2 m − z − α 2 n + β 2 m = ∑ j = 1 m m j α 2 n + β 2 j z − α 2 n + β 2 m − j ≤ α 2 n + β 2 ∑ j = 1 m m j α 2 n + β 2 j − 1 z − α 2 n + β 2 m − j ≤ α 2 n + β 2 ∑ j = 1 m − 1 m j + 1 α 2 n + β 2 j r m − 1 − j = α 2 n + β 2 ∑ j = 0 m − 1 m − 1 j m j + 1 α 2 n + β 2 j r m − 1 − j ≤ α 2 n + β 2 m ∑ j = 0 m − 1 m − 1 j α 2 n + β 2 j r m − 1 − j ≤ m α 2 n + β 2 r + α 2 n + β 2 m − 1 ≤ m α 2 n + β 2 r ¯ m − 1 .

Case 2. 1 ≤ n ≤ m . By Lemma 1, we have

∣ U ¯ n , α , β ( e m ; z ) − e m ( z ) ∣ ≤ r ¯ m + z − α 2 n + β 2 + α 2 n + β 2 m ≤ 2 r ¯ m ≤ 2 m ( m + β 2 ) n r ¯ m .

In conclusion, we obtain that

∣ U ¯ n , α , β ( f ; z ) − f ( z ) ∣ ≤ K r , n ( α , β ) ( f ) .

Denoting by Γ the circle of center z 0 ≔ α 2 n + β 2 , and radius r 1 > r . Since for any z − α 2 n + β 2 ≤ r , v ∈ Γ , we have ∣ v − z ∣ ≥ r 1 − r . By Cauchy’s formulas, we have for z − α 2 n + β 2 ≤ r , n , p ∈ N , that

∣ U ¯ n , α , β ( f ; z ) ( p ) − f ( z ) ( p ) ∣ = p ! 2 π ∫ Γ U ¯ n , α , β ( f ; z ) − f ( z ) ( v − z ) p + 1 d v ≤ K r 1 , n ( α , β ) ( f ) p ! 2 π 2 π r 1 ( r 1 − 1 ) p + 1 = K r 1 , n ( α , β ) ( f ) p ! r 1 ( r 1 − r ) p + 1 .

Acknowledgement

The authors express their sincere gratitude to the anonymous referees for their careful reading of the manuscript and valuable comments.

  1. Funding information: Research of the first author was supported by the National Natural Science Foundation of China under Grant no 12271133.

  2. Author contributions: All authors contributed equally to this work.

  3. Conflict of interest: The authors state that there is no conflict of interest.

  4. Ethical approval: Not applicable.

  5. Data availability statement: Not applicable.

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Received: 2024-05-26
Revised: 2024-10-06
Accepted: 2024-10-15
Published Online: 2025-01-21

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Research Articles
  2. On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
  3. Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
  4. Grand Triebel-Lizorkin-Morrey spaces
  5. Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
  6. Proofs of two conjectures involving sums of normalized Narayana numbers
  7. On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
  8. New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
  9. Some inequalities for rational function with prescribed poles and restricted zeros
  10. Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
  11. Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
  12. Fixed point results for contractions of polynomial type
  13. Estimation for spatial semi-functional partial linear regression model with missing response at random
  14. Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
  15. New forms of bilateral inequalities for K-g-frames
  16. Rate of pole detection using Padé approximants to polynomial expansions
  17. Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
  18. Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
  19. Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
  20. New topologies derived from the old one via operators
  21. Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
  22. Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
  23. Nonparametric expectile shortfall regression for functional data
  24. Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
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  30. Solutions of several general quadratic partial differential-difference equations in ℂ2
  31. Inequalities for the generalized trigonometric functions with respect to weighted power mean
  32. Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
  33. Hankel determinants for q-starlike functions connected with q-sine function
  34. System of partial differential hemivariational inequalities involving nonlocal boundary conditions
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  36. A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
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  38. Behavior of spatial curves under different transformations in Euclidean 4-space
  39. Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
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  41. A study of generalized Mittag-Leffler-type function of arbitrary order
  42. On the approximation of Kantorovich-type Szàsz-Charlier operators
  43. Split quaternion Fourier transforms for two-dimensional real invariant field
  44. Review Article
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  46. Special Issue on Differential Equations and Numerical Analysis - Part II
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  48. Persistence of a unique periodic wave train in convecting shallow water fluid
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  69. Faster approximation to multivariate functions by combined Bernstein-Taylor operators
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  71. Some special functions and cylindrical diffusion equation on α-time scale
  72. (q, p)-Mixing Bloch maps
  73. Orthogonalizing q-Bernoulli polynomials
  74. On better approximation order for the max-product Meyer-König and Zeller operator
  75. Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
  76. Special Issue on Variational Methods and Nonlinear PDEs
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https://doi.org/10.1515/dema-2024-0092
Keywords for this article
complex Bernstein-Durrmeyer-type operators; movable compact disks; approximation rates
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
  3. Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
  4. Grand Triebel-Lizorkin-Morrey spaces
  5. Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
  6. Proofs of two conjectures involving sums of normalized Narayana numbers
  7. On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
  8. New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
  9. Some inequalities for rational function with prescribed poles and restricted zeros
  10. Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
  11. Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
  12. Fixed point results for contractions of polynomial type
  13. Estimation for spatial semi-functional partial linear regression model with missing response at random
  14. Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
  15. New forms of bilateral inequalities for K-g-frames
  16. Rate of pole detection using Padé approximants to polynomial expansions
  17. Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
  18. Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
  19. Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
  20. New topologies derived from the old one via operators
  21. Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
  22. Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
  23. Nonparametric expectile shortfall regression for functional data
  24. Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
  25. A generalized p-Laplacian problem with parameters
  26. A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
  27. Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
  28. ω ℒ -biprojective and ω ¯ -contractible Banach algebras
  29. Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
  30. Solutions of several general quadratic partial differential-difference equations in ℂ2
  31. Inequalities for the generalized trigonometric functions with respect to weighted power mean
  32. Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
  33. Hankel determinants for q-starlike functions connected with q-sine function
  34. System of partial differential hemivariational inequalities involving nonlocal boundary conditions
  35. A new family of multivalent functions defined by certain forms of the quantum integral operator
  36. A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
  37. Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
  38. Behavior of spatial curves under different transformations in Euclidean 4-space
  39. Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
  40. A new kind of Durrmeyer-Stancu-type operators
  41. A study of generalized Mittag-Leffler-type function of arbitrary order
  42. On the approximation of Kantorovich-type Szàsz-Charlier operators
  43. Split quaternion Fourier transforms for two-dimensional real invariant field
  44. Review Article
  45. Characterization generalized derivations of tensor products of nonassociative algebras
  46. Special Issue on Differential Equations and Numerical Analysis - Part II
  47. Existence and optimal control of Hilfer fractional evolution equations
  48. Persistence of a unique periodic wave train in convecting shallow water fluid
  49. Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
  50. Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
  51. Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
  52. Stability and bifurcation analysis of a modified chemostat model
  53. Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
  54. Analytic solutions of a generalized complex multi-dimensional system with fractional order
  55. Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
  56. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
  57. Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
  58. On the sum form functional equation related to diversity index
  59. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
  60. Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
  61. Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
  62. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
  63. Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
  64. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
  65. Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
  66. New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
  67. Special Issue on Approximation Theory and Special Functions 2024 conference
  68. Ulam-type stability for Caputo-type fractional delay differential equations
  69. Faster approximation to multivariate functions by combined Bernstein-Taylor operators
  70. (λ, ψ)-Bernstein-Kantorovich operators
  71. Some special functions and cylindrical diffusion equation on α-time scale
  72. (q, p)-Mixing Bloch maps
  73. Orthogonalizing q-Bernoulli polynomials
  74. On better approximation order for the max-product Meyer-König and Zeller operator
  75. Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
  76. Special Issue on Variational Methods and Nonlinear PDEs
  77. A note on mean field type equations
  78. Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
  79. Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
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